**Identifier**

**1527 statistics**in the database. There are possibly some more waiting for verification.

**Alternating sign matrices**(23 statistics) # matrix like objects

St000065Alternating sign matrices ⟶ ℤ

The number of entries equal to -1 in an alternating sign matrix.

St000066Alternating sign matrices ⟶ ℤ

The column of the unique '1' in the first row of the alternating sign matrix.

St000067Alternating sign matrices ⟶ ℤ

The inversion number of the alternating sign matrix.

St000076Alternating sign matrices ⟶ ℤ

The rank of the alternating sign matrix in the alternating sign matrix poset.

St000134Alternating sign matrices ⟶ ℤ

The size of the orbit of an alternating sign matrix under gyration.

St000187Alternating sign matrices ⟶ ℤ

The determinant of an alternating sign matrix.

St000193Alternating sign matrices ⟶ ℤ

The row of the unique '1' in the first column of the alternating sign matrix.

St000197Alternating sign matrices ⟶ ℤ

The number of entries equal to positive one in the alternating sign matrix.

St000199Alternating sign matrices ⟶ ℤ

The column of the unique '1' in the last row of the alternating sign matrix.

St000200Alternating sign matrices ⟶ ℤ

The row of the unique '1' in the last column of the alternating sign matrix.

St000227Alternating sign matrices ⟶ ℤ

The osculating paths major index of an alternating sign matrix.

St000332Alternating sign matrices ⟶ ℤ

The positive inversions of an alternating sign matrix.

St000888Alternating sign matrices ⟶ ℤ

The maximal sum of entries on a diagonal of an alternating sign matrix.

St000889Alternating sign matrices ⟶ ℤ

The number of alternating sign matrices with the same antidiagonal sums.

St000890Alternating sign matrices ⟶ ℤ

The number of nonzero entries in an alternating sign matrix.

St000892Alternating sign matrices ⟶ ℤ

The maximal number of nonzero entries on a diagonal of an alternating sign matrix.

St000893Alternating sign matrices ⟶ ℤ

The number of distinct diagonal sums of an alternating sign matrix.

St000894Alternating sign matrices ⟶ ℤ

The trace of an alternating sign matrix.

St000895Alternating sign matrices ⟶ ℤ

The number of ones on the main diagonal of an alternating sign matrix.

St000896Alternating sign matrices ⟶ ℤ

The number of zeros on the main diagonal of an alternating sign matrix.

St000898Alternating sign matrices ⟶ ℤ

The number of maximal entries in the last diagonal of the monotone triangle.

St001030Alternating sign matrices ⟶ ℤ

Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix.

St001260Alternating sign matrices ⟶ ℤ

The permanent of an alternating sign matrix.

**Binary trees**(37 statistics) # tree like structures # Catalan objects # graph like objects

St000045Binary trees ⟶ ℤ

The number of linear extensions of a binary tree.

St000050Binary trees ⟶ ℤ

The depth or height of a binary tree.

St000051Binary trees ⟶ ℤ

The size of the left subtree of a binary tree.

St000061Binary trees ⟶ ℤ

The number of nodes on the left branch of a binary tree.

St000082Binary trees ⟶ ℤ

The number of elements smaller than a binary tree in Tamari order.

St000083Binary trees ⟶ ℤ

The number of left oriented leafs of a binary tree except the first one.

St000118Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree.

St000121Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree.

St000122Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree.

St000125Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[[[.,.],.],.

St000126Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree.

St000127Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree.

St000128Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree.

St000129Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree.

St000130Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree.

St000131Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],.

St000132Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree.

St000161Binary trees ⟶ ℤ

The sum of the sizes of the right subtrees of a binary tree.

St000196Binary trees ⟶ ℤ

The number of occurrences of the contiguous pattern [[.,.],[.,.

St000198Binary trees ⟶ ℤ

A decimal representation of a binary tree as a code word.

St000201Binary trees ⟶ ℤ

The number of leaf nodes in a binary tree.

St000203Binary trees ⟶ ℤ

The number of external nodes of a binary tree.

St000204Binary trees ⟶ ℤ

The number of internal nodes of a binary tree.

St000252Binary trees ⟶ ℤ

The number of nodes of degree 3 of a binary tree.

St000385Binary trees ⟶ ℤ

The number of vertices with out-degree 1 in a binary tree.

St000396Binary trees ⟶ ℤ

The register function (or Horton-Strahler number) of a binary tree.

St000398Binary trees ⟶ ℤ

The sum of the depths of the vertices (or total internal path length) of a binary tree.

St000399Binary trees ⟶ ℤ

The external path length of a binary tree.

St000409Binary trees ⟶ ℤ

The number of pitchforks in a binary tree.

St000411Binary trees ⟶ ℤ

The tree factorial of a binary tree.

St000412Binary trees ⟶ ℤ

The number of binary trees with the same underlying unordered tree.

St000414Binary trees ⟶ ℤ

The binary logarithm of the number of binary trees with the same underlying unordered tree.

St000568Binary trees ⟶ ℤ

The hook number of a binary tree.

St000569Binary trees ⟶ ℤ

The sum of the heights of the vertices of a binary tree.

St000701Binary trees ⟶ ℤ

The protection number of a binary tree.

St000919Binary trees ⟶ ℤ

The number of maximal left branches of a binary tree.

St001376Binary trees ⟶ ℤ

The Colless index of a binary tree.

**Binary words**(65 statistics) # word like objects # path like objects

St000288Binary words ⟶ ℤ

The number of ones in a binary word.

St000289Binary words ⟶ ℤ

The decimal representation of a binary word.

St000290Binary words ⟶ ℤ

The major index of a binary word.

St000291Binary words ⟶ ℤ

The number of descents of a binary word.

St000292Binary words ⟶ ℤ

The number of ascents of a binary word.

St000293Binary words ⟶ ℤ

The number of inversions of a binary word.

St000294Binary words ⟶ ℤ

The number of distinct factors of a binary word.

St000295Binary words ⟶ ℤ

The length of the border of a binary word.

St000296Binary words ⟶ ℤ

The length of the symmetric border of a binary word.

St000297Binary words ⟶ ℤ

The number of leading ones in a binary word.

St000326Binary words ⟶ ℤ

The position of the first one in a binary word after appending a 1 at the end.

St000347Binary words ⟶ ℤ

The inversion sum of a binary word.

St000348Binary words ⟶ ℤ

The non-inversion sum of a binary word.

St000389Binary words ⟶ ℤ

The number of runs of ones of odd length in a binary word.

St000390Binary words ⟶ ℤ

The number of runs of ones in a binary word.

St000391Binary words ⟶ ℤ

The sum of the positions of the ones in a binary word.

St000392Binary words ⟶ ℤ

The length of the longest run of ones in a binary word.

St000393Binary words ⟶ ℤ

The number of strictly increasing runs in a binary word.

St000518Binary words ⟶ ℤ

The number of distinct subsequences in a binary word.

St000519Binary words ⟶ ℤ

The largest length of a factor maximising the subword complexity.

St000529Binary words ⟶ ℤ

The number of permutations whose descent word is the given binary word.

St000543Binary words ⟶ ℤ

The size of the conjugacy class of a binary word.

St000626Binary words ⟶ ℤ

The minimal period of a binary word.

St000627Binary words ⟶ ℤ

The exponent of a binary word.

St000628Binary words ⟶ ℤ

The balance of a binary word.

St000629Binary words ⟶ ℤ

The defect of a binary word.

St000630Binary words ⟶ ℤ

The length of the shortest palindromic decomposition of a binary word.

St000631Binary words ⟶ ℤ

The number of distinct palindromic decompositions of a binary word.

St000682Binary words ⟶ ℤ

The Grundy value of Welter's game on a binary word.

St000691Binary words ⟶ ℤ

The number of changes of a binary word.

St000753Binary words ⟶ ℤ

The Grundy value for the game of Kayles on a binary word.

St000792Binary words ⟶ ℤ

The Grundy value for the game of ruler on a binary word.

St000826Binary words ⟶ ℤ

The stopping time of the decimal representation of the binary word for the 3x+1 problem.

St000827Binary words ⟶ ℤ

The decimal representation of a binary word with a leading 1.

St000847Binary words ⟶ ℤ

The number of standard Young tableaux whose descent set is the binary word.

St000875Binary words ⟶ ℤ

The semilength of the longest Dyck word in the Catalan factorisation of a binary word.

St000876Binary words ⟶ ℤ

The number of factors in the Catalan decomposition of a binary word.

St000877Binary words ⟶ ℤ

The depth of the binary word interpreted as a path.

St000878Binary words ⟶ ℤ

The number of ones minus the number of zeros of a binary word.

St000885Binary words ⟶ ℤ

The number of critical steps in the Catalan decomposition of a binary word.

St000921Binary words ⟶ ℤ

The number of internal inversions of a binary word.

St000922Binary words ⟶ ℤ

The minimal number such that all substrings of this length are unique.

St000982Binary words ⟶ ℤ

The length of the longest constant subword.

St000983Binary words ⟶ ℤ

The length of the longest alternating subword.

St001267Binary words ⟶ ℤ

The length of the Lyndon factorization of the binary word.

St001313Binary words ⟶ ℤ

The number of Dyck paths above the lattice path given by a binary word.

St001355Binary words ⟶ ℤ

Number of non-empty prefixes of a binary word that contain equally many 0's and 1's.

St001365Binary words ⟶ ℤ

The number of lattice paths of the same length weakly above the path given by a binary word.

St001371Binary words ⟶ ℤ

The length of the longest Yamanouchi prefix of a binary word.

St001372Binary words ⟶ ℤ

The length of a longest cyclic run of ones of a binary word.

St001413Binary words ⟶ ℤ

Half the length of the longest even length palindromic prefix of a binary word.

St001414Binary words ⟶ ℤ

Half the length of the longest odd length palindromic prefix of a binary word.

St001415Binary words ⟶ ℤ

The length of the longest palindromic prefix of a binary word.

St001416Binary words ⟶ ℤ

The length of a longest palindromic factor of a binary word.

St001417Binary words ⟶ ℤ

The length of a longest palindromic subword of a binary word.

St001419Binary words ⟶ ℤ

The length of the longest palindromic factor beginning with a one of a binary word.

St001420Binary words ⟶ ℤ

Half the length of a longest factor which is its own reverse-complement of a binary word.

St001421Binary words ⟶ ℤ

Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.

St001423Binary words ⟶ ℤ

The number of distinct cubes in a binary word.

St001424Binary words ⟶ ℤ

The number of distinct squares in a binary word.

St001436Binary words ⟶ ℤ

The index of a given binary word in the lex-order among all its cyclic shifts.

St001437Binary words ⟶ ℤ

The flex of a binary word.

St001485Binary words ⟶ ℤ

The modular major index of a binary word.

St001491Binary words ⟶ ℤ

The number of indecomposable projective-injective modules in the algebra corresponding to a subset.

St001524Binary words ⟶ ℤ

The degree of symmetry of a binary word.

**Cores**(5 statistics) # partition like objects

**Decorated permutations**(2 statistics) # word like objects

**Dyck paths**(267 statistics) # Catalan objects # path like objects

St000005Dyck paths ⟶ ℤ

The bounce statistic of a Dyck path.

St000006Dyck paths ⟶ ℤ

The dinv of a Dyck path.

St000011Dyck paths ⟶ ℤ

The number of touch points (or returns) of a Dyck path.

St000012Dyck paths ⟶ ℤ

The area of a Dyck path.

St000013Dyck paths ⟶ ℤ

The height of a Dyck path.

St000014Dyck paths ⟶ ℤ

The number of parking functions supported by a Dyck path.

St000015Dyck paths ⟶ ℤ

The number of peaks of a Dyck path.

St000024Dyck paths ⟶ ℤ

The number of double up and double down steps of a Dyck path.

St000025Dyck paths ⟶ ℤ

The number of initial rises of a Dyck path.

St000026Dyck paths ⟶ ℤ

The position of the first return of a Dyck path.

St000027Dyck paths ⟶ ℤ

The major index of a Dyck path.

St000032Dyck paths ⟶ ℤ

The number of elements smaller than the given Dyck path in the Tamari Order.

St000038Dyck paths ⟶ ℤ

The product of the heights of the descending steps of a Dyck path.

St000052Dyck paths ⟶ ℤ

The number of valleys of a Dyck path not on the x-axis.

St000053Dyck paths ⟶ ℤ

The number of valleys of the Dyck path.

St000079Dyck paths ⟶ ℤ

The number of alternating sign matrices for a given Dyck path.

St000117Dyck paths ⟶ ℤ

The number of centered tunnels of a Dyck path.

St000120Dyck paths ⟶ ℤ

The number of left tunnels of a Dyck path.

St000144Dyck paths ⟶ ℤ

The pyramid weight of the Dyck path.

St000306Dyck paths ⟶ ℤ

The bounce count of a Dyck path.

St000329Dyck paths ⟶ ℤ

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

St000331Dyck paths ⟶ ℤ

The number of upper interactions of a Dyck path.

St000335Dyck paths ⟶ ℤ

The difference of lower and upper interactions.

St000340Dyck paths ⟶ ℤ

The number of non-final maximal constant sub-paths of length greater than one.

St000369Dyck paths ⟶ ℤ

The dinv deficit of a Dyck path.

St000376Dyck paths ⟶ ℤ

The bounce deficit of a Dyck path.

St000386Dyck paths ⟶ ℤ

The number of factors DDU in a Dyck path.

St000394Dyck paths ⟶ ℤ

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

St000395Dyck paths ⟶ ℤ

The sum of the heights of the peaks of a Dyck path.

St000418Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly below a Dyck path.

St000419Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly above the Dyck path, except for the path itself.

St000420Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly above a Dyck path.

St000421Dyck paths ⟶ ℤ

The number of Dyck paths that are weakly below a Dyck path, except for the path itself.

St000438Dyck paths ⟶ ℤ

The position of the last up step in a Dyck path.

St000439Dyck paths ⟶ ℤ

The position of the first down step of a Dyck path.

St000442Dyck paths ⟶ ℤ

The maximal area to the right of an up step of a Dyck path.

St000443Dyck paths ⟶ ℤ

The number of long tunnels of a Dyck path.

St000444Dyck paths ⟶ ℤ

The length of the maximal rise of a Dyck path.

St000445Dyck paths ⟶ ℤ

The number of rises of length 1 of a Dyck path.

St000476Dyck paths ⟶ ℤ

The sum of the semi-lengths of tunnels before a valley of a Dyck path.

St000617Dyck paths ⟶ ℤ

The number of global maxima of a Dyck path.

St000645Dyck paths ⟶ ℤ

The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.

St000655Dyck paths ⟶ ℤ

The length of the minimal rise of a Dyck path.

St000658Dyck paths ⟶ ℤ

The number of rises of length 2 of a Dyck path.

St000659Dyck paths ⟶ ℤ

The number of rises of length at least 2 of a Dyck path.

St000660Dyck paths ⟶ ℤ

The number of rises of length at least 3 of a Dyck path.

St000661Dyck paths ⟶ ℤ

The number of rises of length 3 of a Dyck path.

St000674Dyck paths ⟶ ℤ

The number of hills of a Dyck path.

St000675Dyck paths ⟶ ℤ

The number of centered multitunnels of a Dyck path.

St000676Dyck paths ⟶ ℤ

The number of odd rises of a Dyck path.

St000678Dyck paths ⟶ ℤ

The number of up steps after the last double rise of a Dyck path.

St000683Dyck paths ⟶ ℤ

The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.

St000684Dyck paths ⟶ ℤ

The global dimension of the LNakayama algebra associated to a Dyck path.

St000685Dyck paths ⟶ ℤ

The dominant dimension of the LNakayama algebra associated to a Dyck path.

St000686Dyck paths ⟶ ℤ

The finitistic dominant dimension of a Dyck path.

St000687Dyck paths ⟶ ℤ

The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.

St000688Dyck paths ⟶ ℤ

The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path.

St000689Dyck paths ⟶ ℤ

The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.

St000790Dyck paths ⟶ ℤ

The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path.

St000791Dyck paths ⟶ ℤ

The number of pairs of left tunnels, one strictly containing the other, of a Dyck path.

St000874Dyck paths ⟶ ℤ

The position of the last double rise in a Dyck path.

St000920Dyck paths ⟶ ℤ

The logarithmic height of a Dyck path.

St000930Dyck paths ⟶ ℤ

The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver.

St000931Dyck paths ⟶ ℤ

The number of occurrences of the pattern UUU in a Dyck path.

St000932Dyck paths ⟶ ℤ

The number of occurrences of the pattern UDU in a Dyck path.

St000946Dyck paths ⟶ ℤ

The sum of the skew hook positions in a Dyck path.

St000947Dyck paths ⟶ ℤ

The major index east count of a Dyck path.

St000949Dyck paths ⟶ ℤ

Gives the number of generalised tilting modules of the corresponding LNakayama algebra.

St000950Dyck paths ⟶ ℤ

Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1.

St000951Dyck paths ⟶ ℤ

The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra.

St000952Dyck paths ⟶ ℤ

Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers.

St000953Dyck paths ⟶ ℤ

The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers.

St000954Dyck paths ⟶ ℤ

Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$.

St000955Dyck paths ⟶ ℤ

Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.

St000964Dyck paths ⟶ ℤ

Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra.

St000965Dyck paths ⟶ ℤ

The sum of the dimension of Ext^i(D(A),A) for i=1,.

St000966Dyck paths ⟶ ℤ

Number of peaks minus the global dimension of the corresponding LNakayama algebra.

St000967Dyck paths ⟶ ℤ

The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra.

St000968Dyck paths ⟶ ℤ

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nâˆ’1}]$ by adding $c_0$ to $c_{nâˆ’1}$.

St000969Dyck paths ⟶ ℤ

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$.

St000970Dyck paths ⟶ ℤ

Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.

St000976Dyck paths ⟶ ℤ

The sum of the positions of double up-steps of a Dyck path.

St000977Dyck paths ⟶ ℤ

MacMahon's equal index of a Dyck path.

St000978Dyck paths ⟶ ℤ

The sum of the positions of double down-steps of a Dyck path.

St000979Dyck paths ⟶ ℤ

Half of MacMahon's equal index of a Dyck path.

St000980Dyck paths ⟶ ℤ

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.

St000981Dyck paths ⟶ ℤ

The length of the longest zigzag subpath.

St000984Dyck paths ⟶ ℤ

The number of boxes below precisely one peak.

St000998Dyck paths ⟶ ℤ

Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

St000999Dyck paths ⟶ ℤ

Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.

St001000Dyck paths ⟶ ℤ

Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.

St001001Dyck paths ⟶ ℤ

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

St001002Dyck paths ⟶ ℤ

Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

St001003Dyck paths ⟶ ℤ

The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

St001006Dyck paths ⟶ ℤ

Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

St001007Dyck paths ⟶ ℤ

Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

St001008Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

St001009Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.

St001010Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.

St001011Dyck paths ⟶ ℤ

Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.

St001012Dyck paths ⟶ ℤ

Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.

St001013Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.

St001014Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path.

St001015Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path.

St001016Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

St001017Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path.

St001018Dyck paths ⟶ ℤ

Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.

St001019Dyck paths ⟶ ℤ

Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

St001020Dyck paths ⟶ ℤ

Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.

St001021Dyck paths ⟶ ℤ

Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path.

St001022Dyck paths ⟶ ℤ

Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.

St001023Dyck paths ⟶ ℤ

Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path.

St001024Dyck paths ⟶ ℤ

Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

St001025Dyck paths ⟶ ℤ

Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path.

St001026Dyck paths ⟶ ℤ

The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path.

St001027Dyck paths ⟶ ℤ

Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path.

St001028Dyck paths ⟶ ℤ

Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

St001031Dyck paths ⟶ ℤ

The height of the bicoloured Motzkin path associated with the Dyck path.

St001032Dyck paths ⟶ ℤ

The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path.

St001033Dyck paths ⟶ ℤ

The normalized area of the parallelogram polyomino associated with the Dyck path.

St001034Dyck paths ⟶ ℤ

The area of the parallelogram polyomino associated with the Dyck path.

St001035Dyck paths ⟶ ℤ

The convexity degree of the parallelogram polyomino associated with the Dyck path.

St001036Dyck paths ⟶ ℤ

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

St001037Dyck paths ⟶ ℤ

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

St001038Dyck paths ⟶ ℤ

The minimal height of a column in the parallelogram polyomino associated with the Dyck path.

St001039Dyck paths ⟶ ℤ

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

St001063Dyck paths ⟶ ℤ

Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.

St001064Dyck paths ⟶ ℤ

Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.

St001065Dyck paths ⟶ ℤ

Number of indecomposable reflexive modules in the corresponding Nakayama algebra.

St001066Dyck paths ⟶ ℤ

The number of simple reflexive modules in the corresponding Nakayama algebra.

St001067Dyck paths ⟶ ℤ

The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.

St001068Dyck paths ⟶ ℤ

Number of torsionless simple modules in the corresponding Nakayama algebra.

St001088Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.

St001089Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.

St001104Dyck paths ⟶ ℤ

The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group.

St001107Dyck paths ⟶ ℤ

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.

St001113Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.

St001125Dyck paths ⟶ ℤ

The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.

St001126Dyck paths ⟶ ℤ

Number of simple module that are 1-regular in the corresponding Nakayama algebra.

St001135Dyck paths ⟶ ℤ

The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.

St001137Dyck paths ⟶ ℤ

Number of simple modules that are 3-regular in the corresponding Nakayama algebra.

St001138Dyck paths ⟶ ℤ

The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra.

St001139Dyck paths ⟶ ℤ

The number of occurrences of hills of size 2 in a Dyck path.

St001140Dyck paths ⟶ ℤ

Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra.

St001141Dyck paths ⟶ ℤ

The number of occurrences of hills of size 3 in a Dyck path.

St001142Dyck paths ⟶ ℤ

The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.

St001159Dyck paths ⟶ ℤ

Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.

St001161Dyck paths ⟶ ℤ

The major index north count of a Dyck path.

St001163Dyck paths ⟶ ℤ

The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra.

St001164Dyck paths ⟶ ℤ

Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules.

St001165Dyck paths ⟶ ℤ

Number of simple modules with even projective dimension in the corresponding Nakayama algebra.

St001166Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra.

St001167Dyck paths ⟶ ℤ

The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.

St001169Dyck paths ⟶ ℤ

Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.

St001170Dyck paths ⟶ ℤ

Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra.

St001172Dyck paths ⟶ ℤ

The number of 1-rises at odd height of a Dyck path.

St001179Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra.

St001180Dyck paths ⟶ ℤ

Number of indecomposable injective modules with projective dimension at most 1.

St001181Dyck paths ⟶ ℤ

Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.

St001182Dyck paths ⟶ ℤ

Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.

St001183Dyck paths ⟶ ℤ

The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.

St001184Dyck paths ⟶ ℤ

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

St001185Dyck paths ⟶ ℤ

The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.

St001186Dyck paths ⟶ ℤ

Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.

St001187Dyck paths ⟶ ℤ

The number of simple modules with grade at least one in the corresponding Nakayama algebra.

St001188Dyck paths ⟶ ℤ

The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path.

St001189Dyck paths ⟶ ℤ

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

St001190Dyck paths ⟶ ℤ

Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra.

St001191Dyck paths ⟶ ℤ

Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$.

St001192Dyck paths ⟶ ℤ

The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.

St001193Dyck paths ⟶ ℤ

The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module.

St001194Dyck paths ⟶ ℤ

The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module

St001195Dyck paths ⟶ ℤ

The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.

St001196Dyck paths ⟶ ℤ

The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.

St001197Dyck paths ⟶ ℤ

The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001198Dyck paths ⟶ ℤ

The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001199Dyck paths ⟶ ℤ

The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001200Dyck paths ⟶ ℤ

The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001201Dyck paths ⟶ ℤ

The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.

St001202Dyck paths ⟶ ℤ

Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nâˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra.

St001203Dyck paths ⟶ ℤ

We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:

St001204Dyck paths ⟶ ℤ

Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nâˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra.

St001205Dyck paths ⟶ ℤ

The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

St001206Dyck paths ⟶ ℤ

The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.

St001210Dyck paths ⟶ ℤ

Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.

St001211Dyck paths ⟶ ℤ

The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.

St001212Dyck paths ⟶ ℤ

The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.

St001213Dyck paths ⟶ ℤ

The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.

St001215Dyck paths ⟶ ℤ

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.

St001216Dyck paths ⟶ ℤ

The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module.

St001217Dyck paths ⟶ ℤ

The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.

St001218Dyck paths ⟶ ℤ

Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1.

St001219Dyck paths ⟶ ℤ

Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive.

St001221Dyck paths ⟶ ℤ

The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module.

St001222Dyck paths ⟶ ℤ

Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module.

St001223Dyck paths ⟶ ℤ

Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.

St001224Dyck paths ⟶ ℤ

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.

St001225Dyck paths ⟶ ℤ

The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.

St001226Dyck paths ⟶ ℤ

The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra.

St001227Dyck paths ⟶ ℤ

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

St001228Dyck paths ⟶ ℤ

The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.

St001229Dyck paths ⟶ ℤ

The vector space dimension of the first extension group between the Jacobson radical J and J^2.

St001230Dyck paths ⟶ ℤ

The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.

St001231Dyck paths ⟶ ℤ

The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.

St001232Dyck paths ⟶ ℤ

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

St001233Dyck paths ⟶ ℤ

The number of indecomposable 2-dimensional modules with projective dimension one.

St001234Dyck paths ⟶ ℤ

The number of indecomposable three dimensional modules with projective dimension one.

St001237Dyck paths ⟶ ℤ

The number of simple modules with injective dimension at most one or dominant dimension at least one.

St001238Dyck paths ⟶ ℤ

The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S.

St001239Dyck paths ⟶ ℤ

The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.

St001240Dyck paths ⟶ ℤ

The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra

St001241Dyck paths ⟶ ℤ

The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one.

St001242Dyck paths ⟶ ℤ

The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path.

St001243Dyck paths ⟶ ℤ

The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path.

St001244Dyck paths ⟶ ℤ

The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.

St001253Dyck paths ⟶ ℤ

The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.

St001254Dyck paths ⟶ ℤ

The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J.

St001255Dyck paths ⟶ ℤ

The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.

St001256Dyck paths ⟶ ℤ

Number of simple reflexive modules that are 2-stable reflexive.

St001257Dyck paths ⟶ ℤ

The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.

St001258Dyck paths ⟶ ℤ

Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.

St001259Dyck paths ⟶ ℤ

The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra.

St001264Dyck paths ⟶ ℤ

The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.

St001265Dyck paths ⟶ ℤ

The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra.

St001266Dyck paths ⟶ ℤ

The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra.

St001273Dyck paths ⟶ ℤ

The projective dimension of the first term in an injective coresolution of the regular module.

St001274Dyck paths ⟶ ℤ

The number of indecomposable injective modules with projective dimension equal to two.

St001275Dyck paths ⟶ ℤ

The projective dimension of the second term in a minimal injective coresolution of the regular module.

St001276Dyck paths ⟶ ℤ

The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.

St001278Dyck paths ⟶ ℤ

The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.

St001289Dyck paths ⟶ ℤ

The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero.

St001290Dyck paths ⟶ ℤ

The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.

St001291Dyck paths ⟶ ℤ

The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.

St001292Dyck paths ⟶ ℤ

The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.

St001294Dyck paths ⟶ ℤ

The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra.

St001295Dyck paths ⟶ ℤ

Gives the vector space dimension of the homomorphism space between J^2 and J^2.

St001296Dyck paths ⟶ ℤ

The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra.

St001297Dyck paths ⟶ ℤ

The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.

St001299Dyck paths ⟶ ℤ

The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra.

St001314Dyck paths ⟶ ℤ

The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.

St001348Dyck paths ⟶ ℤ

The bounce of the parallelogram polyomino associated with the Dyck path.

St001361Dyck paths ⟶ ℤ

The number of lattice paths of the same length that stay weakly above a Dyck path.

St001418Dyck paths ⟶ ℤ

Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.

St001431Dyck paths ⟶ ℤ

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.

St001471Dyck paths ⟶ ℤ

The magnitude of a Dyck path.

St001473Dyck paths ⟶ ℤ

The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra.

St001480Dyck paths ⟶ ℤ

The number of simple summands of the module J^2/J^3.

St001481Dyck paths ⟶ ℤ

The minimal height of a peak of a Dyck path.

St001483Dyck paths ⟶ ℤ

The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.

St001492Dyck paths ⟶ ℤ

The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.

St001493Dyck paths ⟶ ℤ

The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.

St001498Dyck paths ⟶ ℤ

The normalised height of a Nakayama algebra with magnitude 1.

St001499Dyck paths ⟶ ℤ

The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.

St001500Dyck paths ⟶ ℤ

The global dimension of magnitude 1 Nakayama algebras.

St001501Dyck paths ⟶ ℤ

The dominant dimension of magnitude 1 Nakayama algebras.

St001502Dyck paths ⟶ ℤ

The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras.

St001503Dyck paths ⟶ ℤ

The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.

St001504Dyck paths ⟶ ℤ

The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.

St001505Dyck paths ⟶ ℤ

The number of elements generated by the Dyck path as a map in the full transformation monoid.

St001506Dyck paths ⟶ ℤ

Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra.

St001507Dyck paths ⟶ ℤ

The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths.

St001508Dyck paths ⟶ ℤ

The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary.

St001509Dyck paths ⟶ ℤ

The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary.

St001514Dyck paths ⟶ ℤ

The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.

St001515Dyck paths ⟶ ℤ

The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).

St001523Dyck paths ⟶ ℤ

The degree of symmetry of a Dyck path.

St001526Dyck paths ⟶ ℤ

The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.

**Finite Cartan types**(41 statistics) # Cartan type objects

St000106Finite Cartan types ⟶ ℤ

The size of the associated Weyl group.

St000107Finite Cartan types ⟶ ℤ

The dimension of the representation $V(\Lambda_1)$.

St000113Finite Cartan types ⟶ ℤ

The rank of the Cartan type.

St000138Finite Cartan types ⟶ ℤ

The Catalan number of an irreducible finite Cartan type.

St000139Finite Cartan types ⟶ ℤ

The Coxeter number of a finite Cartan type.

St000140Finite Cartan types ⟶ ℤ

The positive Catalan number of an irreducible finite Cartan type.

St000821Finite Cartan types ⟶ ℤ

The determinant of the Cartan matrix.

St000851Finite Cartan types ⟶ ℤ

The third Fuss-Catalan number of a finite Cartan type.

St000852Finite Cartan types ⟶ ℤ

The second Fuss-Catalan number of a finite Cartan type.

St000853Finite Cartan types ⟶ ℤ

The number of almost positive roots of a finite Cartan type.

St000854Finite Cartan types ⟶ ℤ

The number of orbits of reflections of a finite Cartan type.

St000855Finite Cartan types ⟶ ℤ

The number of full-support reflections in the Weyl group of a finite Cartan type.

St000856Finite Cartan types ⟶ ℤ

The number of conjugacy classes in the Weyl group of a finite Cartan type.

St000857Finite Cartan types ⟶ ℤ

The number of reflections of the Weyl group of a finite Cartan type.

St000858Finite Cartan types ⟶ ℤ

The number of factorizations of any Coxeter element into reflections of a finite Cartan type.

St000859Finite Cartan types ⟶ ℤ

The number of parking functions of a finite Cartan type.

St000860Finite Cartan types ⟶ ℤ

The size of the center of the Weyl group of a finite Cartan type.

St000861Finite Cartan types ⟶ ℤ

The maximal dimension of an irreducible representation of the Weyl group of a finite Cartan type.

St000865Finite Cartan types ⟶ ℤ

The number of Coxeter elements in the Weyl group of a finite Cartan type.

St000960Finite Cartan types ⟶ ℤ

The permanent of the Cartan matrix of a finite Cartan type.

St001053Finite Cartan types ⟶ ℤ

The second positive Fuss-Catalan number of a finite Cartan type.

St001054Finite Cartan types ⟶ ℤ

The third positive Fuss-Catalan number of a finite Cartan type.

St001143Finite Cartan types ⟶ ℤ

The number of pairs in the Weyl group of given type with mu-coefficient of the Kazhdan Lusztig polynomial being non-zero.

St001144Finite Cartan types ⟶ ℤ

The largest mu-coefficient of the Kazhdan Lusztig polynomial occurring in the Weyl group of given type.

St001145Finite Cartan types ⟶ ℤ

The largest coefficient in a Kazhdan Lusztig polynomial of the Weyl group of given type.

St001146Finite Cartan types ⟶ ℤ

The number of Grassmannian elements in the Coxeter group of the given type.

St001147Finite Cartan types ⟶ ℤ

The number of minuscule dominant weights in the weight lattice of a finite Cartan type.

St001148Finite Cartan types ⟶ ℤ

The dimension of the adjoint representation of the Lie group of given type.

St001149Finite Cartan types ⟶ ℤ

The dimension of the quasi-minuscule representation of the Lie group of given type.

St001150Finite Cartan types ⟶ ℤ

The minimal dimension of a faithful linear representation of the Lie algebra of given type.

St001154Finite Cartan types ⟶ ℤ

The dual Coxeter number of a finite Cartan type.

St001155Finite Cartan types ⟶ ℤ

The number of conjugacy classes of subgroups of the Weyl group of given type.

St001156Finite Cartan types ⟶ ℤ

The Dynkin index of the Lie algebra of given type.

St001157Finite Cartan types ⟶ ℤ

The exponent of the Weyl group of given type.

St001158Finite Cartan types ⟶ ℤ

The size of the mutation class of quivers of given type.

St001173Finite Cartan types ⟶ ℤ

The number of commutative positive roots in the root system of the given finite Cartan type.

St001369Finite Cartan types ⟶ ℤ

The largest coefficient in the highest root in the root system of a Cartan type.

St001370Finite Cartan types ⟶ ℤ

The degree of the largest fundamental representation associated with a Cartan type.

St001443Finite Cartan types ⟶ ℤ

The largest coefficient in the PoincarÃ© polynomial of the Weyl group of given Cartan type.

St001467Finite Cartan types ⟶ ℤ

The number of involutions in the Weyl group of a given Cartan type.

St001495Finite Cartan types ⟶ ℤ

The maximal order of an element in the Weyl group of a given Cartan type.

**Gelfand-Tsetlin patterns**(13 statistics) # matrix like objects

St000072Gelfand-Tsetlin patterns ⟶ ℤ

The number of circled entries.

St000073Gelfand-Tsetlin patterns ⟶ ℤ

The number of boxed entries.

St000074Gelfand-Tsetlin patterns ⟶ ℤ

The number of special entries.

St000077Gelfand-Tsetlin patterns ⟶ ℤ

The number of boxed and circled entries.

St000114Gelfand-Tsetlin patterns ⟶ ℤ

The sum of the entries of the Gelfand-Tsetlin pattern.

St000115Gelfand-Tsetlin patterns ⟶ ℤ

The single entry in the last row.

St000152Gelfand-Tsetlin patterns ⟶ ℤ

The number of boxed plus the number of special entries.

St000176Gelfand-Tsetlin patterns ⟶ ℤ

The total number of tiles in the Gelfand-Tsetlin pattern.

St000177Gelfand-Tsetlin patterns ⟶ ℤ

The number of free tiles in the pattern.

St000178Gelfand-Tsetlin patterns ⟶ ℤ

Number of free entries.

St000186Gelfand-Tsetlin patterns ⟶ ℤ

The sum of the first row in a Gelfand-Tsetlin pattern.

St001404Gelfand-Tsetlin patterns ⟶ ℤ

The number of distinct entries in a Gelfand Tsetlin pattern.

St001406Gelfand-Tsetlin patterns ⟶ ℤ

The number of nonzero entries in a Gelfand Tsetlin pattern.

**Graphs**(217 statistics) # graph like objects

St000081Graphs ⟶ ℤ

The number of edges of a graph.

St000086Graphs ⟶ ℤ

The number of subgraphs.

St000087Graphs ⟶ ℤ

The number of induced subgraphs.

St000093Graphs ⟶ ℤ

The length of the maximal independent set of vertices of a graph.

St000095Graphs ⟶ ℤ

The number of triangles of a graph.

St000096Graphs ⟶ ℤ

The number of spanning trees of a graph.

St000097Graphs ⟶ ℤ

The order of the largest clique of the graph.

St000098Graphs ⟶ ℤ

The chromatic number of a graph.

St000171Graphs ⟶ ℤ

The degree of the graph.

St000172Graphs ⟶ ℤ

The Grundy number of a graph.

St000244Graphs ⟶ ℤ

The cardinality of the automorphism group of a graph.

St000258Graphs ⟶ ℤ

The burning number of a graph.

St000259Graphs ⟶ ℤ

The diameter of a connected graph.

St000260Graphs ⟶ ℤ

The radius of a connected graph.

St000261Graphs ⟶ ℤ

The edge connectivity of a graph.

St000262Graphs ⟶ ℤ

The vertex connectivity of a graph.

St000263Graphs ⟶ ℤ

The Szeged index of a graph.

St000264Graphs ⟶ ℤ

The girth of a graph, which is not a tree.

St000265Graphs ⟶ ℤ

The Wiener index of a graph.

St000266Graphs ⟶ ℤ

The number of spanning subgraphs of a graph with the same connected components.

St000267Graphs ⟶ ℤ

The number of maximal spanning forests contained in a graph.

St000268Graphs ⟶ ℤ

The number of strongly connected orientations of a graph.

St000269Graphs ⟶ ℤ

The number of acyclic orientations of a graph.

St000270Graphs ⟶ ℤ

The number of forests contained in a graph.

St000271Graphs ⟶ ℤ

The chromatic index of a connected graph.

St000272Graphs ⟶ ℤ

The treewidth of a graph.

St000273Graphs ⟶ ℤ

The domination number of a graph.

St000274Graphs ⟶ ℤ

The number of perfect matchings of a graph.

St000276Graphs ⟶ ℤ

The size of the preimage of the map 'to graph' from Ordered trees to Graphs.

St000283Graphs ⟶ ℤ

The size of the preimage of the map 'to graph' from Binary trees to Graphs.

St000286Graphs ⟶ ℤ

The number of connected components of the complement of a graph.

St000287Graphs ⟶ ℤ

The number of connected components of a graph.

St000299Graphs ⟶ ℤ

The number of nonisomorphic vertex-induced subtrees.

St000300Graphs ⟶ ℤ

The number of independent sets of vertices of a graph.

St000301Graphs ⟶ ℤ

The number of facets of the stable set polytope of a graph.

St000302Graphs ⟶ ℤ

The determinant of the distance matrix of a connected graph.

St000303Graphs ⟶ ℤ

The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$.

St000309Graphs ⟶ ℤ

The number of vertices with even degree.

St000310Graphs ⟶ ℤ

The minimal degree of a vertex of a graph.

St000311Graphs ⟶ ℤ

The number of vertices of odd degree in a graph.

St000312Graphs ⟶ ℤ

The number of leaves in a graph.

St000313Graphs ⟶ ℤ

The number of degree 2 vertices of a graph.

St000315Graphs ⟶ ℤ

The number of isolated vertices of a graph.

St000322Graphs ⟶ ℤ

The skewness of a graph.

St000323Graphs ⟶ ℤ

The minimal crossing number of a graph.

St000343Graphs ⟶ ℤ

The number of spanning subgraphs of a graph.

St000344Graphs ⟶ ℤ

The number of strongly connected outdegree sequences of a graph.

St000349Graphs ⟶ ℤ

The number of different adjacency matrices of a graph.

St000350Graphs ⟶ ℤ

The sum of the vertex degrees of a graph.

St000351Graphs ⟶ ℤ

The determinant of the adjacency matrix of a graph.

St000361Graphs ⟶ ℤ

The second Zagreb index of a graph.

St000362Graphs ⟶ ℤ

The size of a minimal vertex cover of a graph.

St000363Graphs ⟶ ℤ

The number of minimal vertex covers of a graph.

St000364Graphs ⟶ ℤ

The exponent of the automorphism group of a graph.

St000368Graphs ⟶ ℤ

The Altshuler-Steinberg determinant of a graph.

St000370Graphs ⟶ ℤ

The genus of a graph.

St000379Graphs ⟶ ℤ

The number of Hamiltonian cycles in a graph.

St000387Graphs ⟶ ℤ

The matching number of a graph.

St000388Graphs ⟶ ℤ

The number of orbits of vertices of a graph under automorphisms.

St000403Graphs ⟶ ℤ

The Szeged index minus the Wiener index of a graph.

St000422Graphs ⟶ ℤ

The energy of a graph, if it is integral.

St000447Graphs ⟶ ℤ

The number of pairs of vertices of a graph with distance 3.

St000448Graphs ⟶ ℤ

The number of pairs of vertices of a graph with distance 2.

St000449Graphs ⟶ ℤ

The number of pairs of vertices of a graph with distance 4.

St000450Graphs ⟶ ℤ

The number of edges minus the number of vertices plus 2 of a graph.

St000452Graphs ⟶ ℤ

The number of distinct eigenvalues of a graph.

St000453Graphs ⟶ ℤ

The number of distinct Laplacian eigenvalues of a graph.

St000454Graphs ⟶ ℤ

The largest eigenvalue of a graph if it is integral.

St000455Graphs ⟶ ℤ

The second largest eigenvalue of a graph if it is integral.

St000456Graphs ⟶ ℤ

The monochromatic index of a connected graph.

St000464Graphs ⟶ ℤ

The Schultz index of a connected graph.

St000465Graphs ⟶ ℤ

The first Zagreb index of a graph.

St000466Graphs ⟶ ℤ

The Gutman (or modified Schultz) index of a connected graph.

St000467Graphs ⟶ ℤ

The hyper-Wiener index of a connected graph.

St000468Graphs ⟶ ℤ

The Hosoya index of a graph.

St000469Graphs ⟶ ℤ

The distinguishing number of a graph.

St000479Graphs ⟶ ℤ

The Ramsey number of a graph.

St000482Graphs ⟶ ℤ

The (zero)-forcing number of a graph.

St000535Graphs ⟶ ℤ

The rank-width of a graph.

St000536Graphs ⟶ ℤ

The pathwidth of a graph.

St000537Graphs ⟶ ℤ

The cutwidth of a graph.

St000544Graphs ⟶ ℤ

The cop number of a graph.

St000552Graphs ⟶ ℤ

The number of cut vertices of a graph.

St000553Graphs ⟶ ℤ

The number of blocks of a graph.

St000571Graphs ⟶ ℤ

The F-index (or forgotten topological index) of a graph.

St000636Graphs ⟶ ℤ

The hull number of a graph.

St000637Graphs ⟶ ℤ

The length of the longest cycle in a graph.

St000671Graphs ⟶ ℤ

The maximin edge-connectivity for choosing a subgraph.

St000699Graphs ⟶ ℤ

The toughness times the least common multiple of 1,.

St000718Graphs ⟶ ℤ

The largest Laplacian eigenvalue of a graph if it is integral.

St000722Graphs ⟶ ℤ

The number of different neighbourhoods in a graph.

St000723Graphs ⟶ ℤ

The maximal cardinality of a set of vertices with the same neighbourhood in a graph.

St000741Graphs ⟶ ℤ

The Colin de VerdiÃ¨re graph invariant.

St000771Graphs ⟶ ℤ

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.

St000772Graphs ⟶ ℤ

The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.

St000773Graphs ⟶ ℤ

The multiplicity of the largest Laplacian eigenvalue in a graph.

St000774Graphs ⟶ ℤ

The maximal multiplicity of a Laplacian eigenvalue in a graph.

St000775Graphs ⟶ ℤ

The multiplicity of the largest eigenvalue in a graph.

St000776Graphs ⟶ ℤ

The maximal multiplicity of an eigenvalue in a graph.

St000777Graphs ⟶ ℤ

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

St000778Graphs ⟶ ℤ

The metric dimension of a graph.

St000785Graphs ⟶ ℤ

The number of distinct colouring schemes of a graph.

St000786Graphs ⟶ ℤ

The maximal number of occurrences of a colour in a proper colouring of a graph.

St000822Graphs ⟶ ℤ

The Hadwiger number of the graph.

St000915Graphs ⟶ ℤ

The Ore degree of a graph.

St000916Graphs ⟶ ℤ

The packing number of a graph.

St000917Graphs ⟶ ℤ

The open packing number of a graph.

St000918Graphs ⟶ ℤ

The 2-limited packing number of a graph.

St000926Graphs ⟶ ℤ

The clique-coclique number of a graph.

St000948Graphs ⟶ ℤ

The chromatic discriminant of a graph.

St000972Graphs ⟶ ℤ

The composition number of a graph.

St000985Graphs ⟶ ℤ

The number of positive eigenvalues of the adjacency matrix of the graph.

St000986Graphs ⟶ ℤ

The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.

St000987Graphs ⟶ ℤ

The number of positive eigenvalues of the Laplacian matrix of the graph.

St001029Graphs ⟶ ℤ

The size of the core of a graph.

St001056Graphs ⟶ ℤ

The Grundy value for the game of deleting vertices of a graph until it has no edges.

St001057Graphs ⟶ ℤ

The Grundy value of the game of creating an independent set in a graph.

St001060Graphs ⟶ ℤ

The distinguishing index of a graph.

St001069Graphs ⟶ ℤ

The coefficient of the monomial xy of the Tutte polynomial of the graph.

St001070Graphs ⟶ ℤ

The absolute value of the derivative of the chromatic polynomial of the graph at 1.

St001071Graphs ⟶ ℤ

The beta invariant of the graph.

St001072Graphs ⟶ ℤ

The evaluation of the Tutte polynomial of the graph at x and y equal to 3.

St001073Graphs ⟶ ℤ

The number of nowhere zero 3-flows of a graph.

St001093Graphs ⟶ ℤ

The detour number of a graph.

St001108Graphs ⟶ ℤ

The 2-dynamic chromatic number of a graph.

St001109Graphs ⟶ ℤ

The number of proper colourings of a graph with as few colours as possible.

St001110Graphs ⟶ ℤ

The 3-dynamic chromatic number of a graph.

St001111Graphs ⟶ ℤ

The weak 2-dynamic chromatic number of a graph.

St001112Graphs ⟶ ℤ

The 3-weak dynamic number of a graph.

St001116Graphs ⟶ ℤ

The game chromatic number of a graph.

St001117Graphs ⟶ ℤ

The game chromatic index of a graph.

St001118Graphs ⟶ ℤ

The acyclic chromatic index of a graph.

St001119Graphs ⟶ ℤ

The length of a shortest maximal path in a graph.

St001120Graphs ⟶ ℤ

The length of a longest path in a graph.

St001261Graphs ⟶ ℤ

The Castelnuovo-Mumford regularity of a graph.

St001270Graphs ⟶ ℤ

The bandwidth of a graph.

St001271Graphs ⟶ ℤ

The competition number of a graph.

St001272Graphs ⟶ ℤ

The number of graphs with the same degree sequence.

St001277Graphs ⟶ ℤ

The degeneracy of a graph.

St001281Graphs ⟶ ℤ

The normalized isoperimetric number of a graph.

St001282Graphs ⟶ ℤ

The number of graphs with the same chromatic polynomial.

St001286Graphs ⟶ ℤ

The annihilation number of a graph.

St001302Graphs ⟶ ℤ

The number of minimally dominating sets of vertices of a graph.

St001303Graphs ⟶ ℤ

The number of dominating sets of vertices of a graph.

St001304Graphs ⟶ ℤ

The number of maximally independent sets of vertices of a graph.

St001305Graphs ⟶ ℤ

The number of induced cycles on four vertices in a graph.

St001306Graphs ⟶ ℤ

The number of induced paths on four vertices in a graph.

St001307Graphs ⟶ ℤ

The number of induced stars on four vertices in a graph.

St001308Graphs ⟶ ℤ

The number of induced paths on three vertices in a graph.

St001309Graphs ⟶ ℤ

The number of four-cliques in a graph.

St001310Graphs ⟶ ℤ

The number of induced diamond graphs in a graph.

St001311Graphs ⟶ ℤ

The cyclomatic number of a graph.

St001315Graphs ⟶ ℤ

The dissociation number of a graph.

St001316Graphs ⟶ ℤ

The domatic number of a graph.

St001317Graphs ⟶ ℤ

The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph.

St001318Graphs ⟶ ℤ

The number of vertices of the largest induced subforest with the same number of connected components of a graph.

St001319Graphs ⟶ ℤ

The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph.

St001320Graphs ⟶ ℤ

The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph.

St001321Graphs ⟶ ℤ

The number of vertices of the largest induced subforest of a graph.

St001322Graphs ⟶ ℤ

The size of a minimal independent dominating set in a graph.

St001323Graphs ⟶ ℤ

The independence gap of a graph.

St001324Graphs ⟶ ℤ

The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph.

St001325Graphs ⟶ ℤ

The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph.

St001326Graphs ⟶ ℤ

The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph.

St001327Graphs ⟶ ℤ

The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph.

St001328Graphs ⟶ ℤ

The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph.

St001329Graphs ⟶ ℤ

The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph.

St001330Graphs ⟶ ℤ

The hat guessing number of a graph.

St001331Graphs ⟶ ℤ

The size of the minimal feedback vertex set.

St001333Graphs ⟶ ℤ

The cardinality of a minimal edge-isolating set of a graph.

St001334Graphs ⟶ ℤ

The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph.

St001335Graphs ⟶ ℤ

The cardinality of a minimal cycle-isolating set of a graph.

St001336Graphs ⟶ ℤ

The minimal number of vertices in a graph whose complement is triangle-free.

St001337Graphs ⟶ ℤ

The upper domination number of a graph.

St001338Graphs ⟶ ℤ

The upper irredundance number of a graph.

St001339Graphs ⟶ ℤ

The irredundance number of a graph.

St001340Graphs ⟶ ℤ

The cardinality of a minimal non-edge isolating set of a graph.

St001341Graphs ⟶ ℤ

The number of edges in the center of a graph.

St001342Graphs ⟶ ℤ

The number of vertices in the center of a graph.

St001345Graphs ⟶ ℤ

The Hamming dimension of a graph.

St001347Graphs ⟶ ℤ

The number of pairs of vertices of a graph having the same neighbourhood.

St001349Graphs ⟶ ℤ

The number of different graphs obtained from the given graph by removing an edge.

St001350Graphs ⟶ ℤ

Half of the Albertson index of a graph.

St001351Graphs ⟶ ℤ

The Albertson index of a graph.

St001352Graphs ⟶ ℤ

The number of internal nodes in the modular decomposition of a graph.

St001353Graphs ⟶ ℤ

The number of prime nodes in the modular decomposition of a graph.

St001354Graphs ⟶ ℤ

The number of series nodes in the modular decomposition of a graph.

St001356Graphs ⟶ ℤ

The number of vertices in prime modules of a graph.

St001357Graphs ⟶ ℤ

The maximal degree of a regular spanning subgraph of a graph.

St001358Graphs ⟶ ℤ

The largest degree of a regular subgraph of a graph.

St001362Graphs ⟶ ℤ

The normalized Knill dimension of a graph.

St001363Graphs ⟶ ℤ

The Euler characteristic of a graph according to Knill.

St001366Graphs ⟶ ℤ

The maximal multiplicity of a degree of a vertex of a graph.

St001367Graphs ⟶ ℤ

The smallest number which does not occur as degree of a vertex in a graph.

St001368Graphs ⟶ ℤ

The number of vertices of maximal degree in a graph.

St001373Graphs ⟶ ℤ

The logarithm of the number of winning configurations of the lights out game on a graph.

St001374Graphs ⟶ ℤ

The Padmakar-Ivan index of a graph.

St001386Graphs ⟶ ℤ

The number of prime labellings of a graph.

St001391Graphs ⟶ ℤ

The disjunction number of a graph.

St001393Graphs ⟶ ℤ

The induced matching number of a graph.

St001395Graphs ⟶ ℤ

The number of strictly unfriendly partitions of a graph.

St001441Graphs ⟶ ℤ

The number of non-empty connected induced subgraphs of a graph.

St001458Graphs ⟶ ℤ

The rank of the adjacency matrix of a graph.

St001459Graphs ⟶ ℤ

The number of zero columns in the nullspace of a graph.

St001463Graphs ⟶ ℤ

The number of distinct columns in the nullspace of a graph.

St001474Graphs ⟶ ℤ

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1).

St001475Graphs ⟶ ℤ

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0).

St001476Graphs ⟶ ℤ

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1).

St001477Graphs ⟶ ℤ

The number of nowhere zero 5-flows of a graph.

St001478Graphs ⟶ ℤ

The number of nowhere zero 4-flows of a graph.

St001479Graphs ⟶ ℤ

The number of bridges of a graph.

St001494Graphs ⟶ ℤ

The Alon-Tarsi number of a graph.

St001496Graphs ⟶ ℤ

The number of graphs with the same Laplacian spectrum as the given graph.

St001512Graphs ⟶ ℤ

The minimum rank of a graph.

St001518Graphs ⟶ ℤ

The number of graphs with the same ordinary spectrum as the given graph.

St001521Graphs ⟶ ℤ

Half the total irregularity of a graph.

St001522Graphs ⟶ ℤ

The total irregularity of a graph.

**Integer compositions**(43 statistics) # partition like objects

St000008Integer compositions ⟶ ℤ

The major index of the composition.

St000047Integer compositions ⟶ ℤ

The number of standard immaculate tableaux of a given shape.

St000089Integer compositions ⟶ ℤ

The absolute variation of a composition.

St000090Integer compositions ⟶ ℤ

The variation of a composition.

St000091Integer compositions ⟶ ℤ

The descent variation of a composition.

St000277Integer compositions ⟶ ℤ

The number of ribbon shaped standard tableaux.

St000285Integer compositions ⟶ ℤ

The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions.

St000381Integer compositions ⟶ ℤ

The largest part of an integer composition.

St000382Integer compositions ⟶ ℤ

The first part of an integer composition.

St000383Integer compositions ⟶ ℤ

The last part of an integer composition.

St000657Integer compositions ⟶ ℤ

The smallest part of an integer composition.

St000757Integer compositions ⟶ ℤ

The length of the longest weakly inreasing subsequence of parts of an integer composition.

St000758Integer compositions ⟶ ℤ

The length of the longest staircase fitting into an integer composition.

St000760Integer compositions ⟶ ℤ

The length of the longest strictly decreasing subsequence of parts of an integer composition.

St000761Integer compositions ⟶ ℤ

The number of ascents in an integer composition.

St000762Integer compositions ⟶ ℤ

The sum of the positions of the weak records of an integer composition.

St000763Integer compositions ⟶ ℤ

The sum of the positions of the strong records of an integer composition.

St000764Integer compositions ⟶ ℤ

The number of strong records in an integer composition.

St000765Integer compositions ⟶ ℤ

The number of weak records in an integer composition.

St000766Integer compositions ⟶ ℤ

The number of inversions of an integer composition.

St000767Integer compositions ⟶ ℤ

The number of runs in an integer composition.

St000768Integer compositions ⟶ ℤ

The number of peaks in an integer composition.

St000769Integer compositions ⟶ ℤ

The major index of a composition.

St000805Integer compositions ⟶ ℤ

The number of peaks of the associated bargraph.

St000806Integer compositions ⟶ ℤ

The semiperimeter of the associated bargraph.

St000807Integer compositions ⟶ ℤ

The sum of the heights of the valleys of the associated bargraph.

St000808Integer compositions ⟶ ℤ

The number of up steps of the associated bargraph.

St000816Integer compositions ⟶ ℤ

The number of standard composition tableaux of the composition.

St000817Integer compositions ⟶ ℤ

The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions.

St000818Integer compositions ⟶ ℤ

The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions.

St000820Integer compositions ⟶ ℤ

The number of compositions obtained by rotating the composition.

St000899Integer compositions ⟶ ℤ

The maximal number of repetitions of an integer composition.

St000900Integer compositions ⟶ ℤ

The minimal number of repetitions of a part in an integer composition.

St000902Integer compositions ⟶ ℤ

The minimal number of repetitions of an integer composition.

St000903Integer compositions ⟶ ℤ

The number of different parts of an integer composition.

St000904Integer compositions ⟶ ℤ

The maximal number of repetitions of an integer composition.

St000905Integer compositions ⟶ ℤ

The number of different multiplicities of parts of an integer composition.

St001102Integer compositions ⟶ ℤ

The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132.

St001235Integer compositions ⟶ ℤ

The global dimension of the corresponding Comp-Nakayama algebra.

St001236Integer compositions ⟶ ℤ

The dominant dimension of the corresponding Comp-Nakayama algebra.

St001263Integer compositions ⟶ ℤ

The index of the maximal parabolic seaweed algebra associated with the composition.

St001312Integer compositions ⟶ ℤ

Number of parabolic noncrossing partitions indexed by the composition.

St001486Integer compositions ⟶ ℤ

The number of corners of the ribbon associated with an integer composition.

**Integer partitions**(180 statistics) # partition like objects

St000003Integer partitions ⟶ ℤ

The number of standard Young tableaux of the partition.

St000010Integer partitions ⟶ ℤ

The length of the partition.

St000046Integer partitions ⟶ ℤ

The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.

St000048Integer partitions ⟶ ℤ

The multinomial of the parts of a partition.

St000049Integer partitions ⟶ ℤ

The number of set partitions whose sorted block sizes correspond to the partition.

St000063Integer partitions ⟶ ℤ

The number of linear extensions of a certain poset defined for an integer partition.

St000088Integer partitions ⟶ ℤ

The row sums of the character table of the symmetric group.

St000108Integer partitions ⟶ ℤ

The number of partitions contained in the given partition.

St000137Integer partitions ⟶ ℤ

The Grundy value of an integer partition.

St000142Integer partitions ⟶ ℤ

The number of even parts of a partition.

St000143Integer partitions ⟶ ℤ

The largest repeated part of a partition.

St000145Integer partitions ⟶ ℤ

The Dyson rank of a partition.

St000146Integer partitions ⟶ ℤ

The Andrews-Garvan crank of a partition.

St000147Integer partitions ⟶ ℤ

The largest part of an integer partition.

St000148Integer partitions ⟶ ℤ

The number of odd parts of a partition.

St000149Integer partitions ⟶ ℤ

The number of cells of the partition whose leg is zero and arm is odd.

St000150Integer partitions ⟶ ℤ

The floored half-sum of the multiplicities of a partition.

St000159Integer partitions ⟶ ℤ

The number of distinct parts of the integer partition.

St000160Integer partitions ⟶ ℤ

The multiplicity of the smallest part of a partition.

St000175Integer partitions ⟶ ℤ

Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.

St000179Integer partitions ⟶ ℤ

The product of the hook lengths of the integer partition.

St000182Integer partitions ⟶ ℤ

The number of permutations whose cycle type is the given integer partition.

St000183Integer partitions ⟶ ℤ

The side length of the Durfee square of an integer partition.

St000184Integer partitions ⟶ ℤ

The size of the centralizer of any permutation of given cycle type.

St000185Integer partitions ⟶ ℤ

The weighted size of a partition.

St000205Integer partitions ⟶ ℤ

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.

St000206Integer partitions ⟶ ℤ

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.

St000207Integer partitions ⟶ ℤ

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.

St000208Integer partitions ⟶ ℤ

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.

St000212Integer partitions ⟶ ℤ

The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row.

St000225Integer partitions ⟶ ℤ

Difference between largest and smallest parts in a partition.

St000228Integer partitions ⟶ ℤ

The size of a partition.

St000256Integer partitions ⟶ ℤ

The number of parts from which one can substract 2 and still get an integer partition.

St000257Integer partitions ⟶ ℤ

The number of distinct parts of a partition that occur at least twice.

St000275Integer partitions ⟶ ℤ

Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition.

St000278Integer partitions ⟶ ℤ

The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.

St000284Integer partitions ⟶ ℤ

The Plancherel distribution on integer partitions.

St000318Integer partitions ⟶ ℤ

The number of addable cells of the Ferrers diagram of an integer partition.

St000319Integer partitions ⟶ ℤ

The spin of an integer partition.

St000320Integer partitions ⟶ ℤ

The dinv adjustment of an integer partition.

St000321Integer partitions ⟶ ℤ

The number of integer partitions of n that are dominated by an integer partition.

St000345Integer partitions ⟶ ℤ

The number of refinements of a partition.

St000346Integer partitions ⟶ ℤ

The number of coarsenings of a partition.

St000377Integer partitions ⟶ ℤ

The dinv defect of an integer partition.

St000378Integer partitions ⟶ ℤ

The diagonal inversion number of an integer partition.

St000380Integer partitions ⟶ ℤ

Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition.

St000384Integer partitions ⟶ ℤ

The maximal part of the shifted composition of an integer partition.

St000459Integer partitions ⟶ ℤ

The hook length of the base cell of a partition.

St000460Integer partitions ⟶ ℤ

The hook length of the last cell along the main diagonal of an integer partition.

St000473Integer partitions ⟶ ℤ

The number of parts of a partition that are strictly bigger than the number of ones.

St000474Integer partitions ⟶ ℤ

Dyson's crank of a partition.

St000475Integer partitions ⟶ ℤ

The number of parts equal to 1 in a partition.

St000477Integer partitions ⟶ ℤ

The weight of a partition according to Alladi.

St000478Integer partitions ⟶ ℤ

Another weight of a partition according to Alladi.

St000480Integer partitions ⟶ ℤ

The number of lower covers of a partition in dominance order.

St000481Integer partitions ⟶ ℤ

The number of upper covers of a partition in dominance order.

St000506Integer partitions ⟶ ℤ

The number of standard desarrangement tableaux of shape equal to the given partition.

St000509Integer partitions ⟶ ℤ

The diagonal index (content) of a partition.

St000510Integer partitions ⟶ ℤ

The number of invariant oriented cycles when acting with a permutation of given cycle type.

St000511Integer partitions ⟶ ℤ

The number of invariant subsets when acting with a permutation of given cycle type.

St000512Integer partitions ⟶ ℤ

The number of invariant subsets of size 3 when acting with a permutation of given cycle type.

St000513Integer partitions ⟶ ℤ

The number of invariant subsets of size 2 when acting with a permutation of given cycle type.

St000514Integer partitions ⟶ ℤ

The number of invariant simple graphs when acting with a permutation of given cycle type.

St000515Integer partitions ⟶ ℤ

The number of invariant set partitions when acting with a permutation of given cycle type.

St000517Integer partitions ⟶ ℤ

The Kreweras number of an integer partition.

St000531Integer partitions ⟶ ℤ

The number of ways to place as many non-attacking rooks as possible on a Ferrers board.

St000532Integer partitions ⟶ ℤ

The total number of rook placements on a Ferrers board.

St000533Integer partitions ⟶ ℤ

The maximal number of non-attacking rooks on a Ferrers shape.

St000547Integer partitions ⟶ ℤ

The number of even non-empty partial sums of an integer partition.

St000548Integer partitions ⟶ ℤ

The number of different non-empty partial sums of an integer partition.

St000549Integer partitions ⟶ ℤ

The number of odd partial sums of an integer partition.

St000566Integer partitions ⟶ ℤ

The number of ways to select a row of a Ferrers shape and two cells in this row.

St000567Integer partitions ⟶ ℤ

The sum of the products of all pairs of parts.

St000618Integer partitions ⟶ ℤ

The number of self-evacuating tableaux of given shape.

St000620Integer partitions ⟶ ℤ

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.

St000621Integer partitions ⟶ ℤ

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.

St000644Integer partitions ⟶ ℤ

The number of graphs with given frequency partition.

St000667Integer partitions ⟶ ℤ

The greatest common divisor of the parts of the partition.

St000668Integer partitions ⟶ ℤ

The least common multiple of the parts of the partition.

St000681Integer partitions ⟶ ℤ

The Grundy value of Chomp on Ferrers diagrams.

St000697Integer partitions ⟶ ℤ

The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core.

St000698Integer partitions ⟶ ℤ

The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.

St000704Integer partitions ⟶ ℤ

The number of semistandard tableaux on a given integer partition with minimal maximal entry.

St000705Integer partitions ⟶ ℤ

The number of semistandard tableaux on a given integer partition of n with maximal entry n.

St000706Integer partitions ⟶ ℤ

The product of the factorials of the multiplicities of an integer partition.

St000707Integer partitions ⟶ ℤ

The product of the factorials of the parts.

St000708Integer partitions ⟶ ℤ

The product of the parts of an integer partition.

St000712Integer partitions ⟶ ℤ

The number of semistandard Young tableau of given shape, with entries at most 4.

St000713Integer partitions ⟶ ℤ

The dimension of the irreducible representation of Sp(4) labelled by an integer partition.

St000714Integer partitions ⟶ ℤ

The number of semistandard Young tableau of given shape, with entries at most 2.

St000715Integer partitions ⟶ ℤ

The number of semistandard Young tableaux of given shape and entries at most 3.

St000716Integer partitions ⟶ ℤ

The dimension of the irreducible representation of Sp(6) labelled by an integer partition.

St000749Integer partitions ⟶ ℤ

The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.

St000752Integer partitions ⟶ ℤ

The Grundy value for the game 'Couples are forever' on an integer partition.

St000755Integer partitions ⟶ ℤ

The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.

St000759Integer partitions ⟶ ℤ

The smallest missing part in an integer partition.

St000770Integer partitions ⟶ ℤ

The major index of an integer partition when read from bottom to top.

St000781Integer partitions ⟶ ℤ

The number of proper colouring schemes of a Ferrers diagram.

St000783Integer partitions ⟶ ℤ

The side length of the largest staircase partition fitting into a partition.

St000784Integer partitions ⟶ ℤ

The maximum of the length and the largest part of the integer partition.

St000810Integer partitions ⟶ ℤ

The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions.

St000811Integer partitions ⟶ ℤ

The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions.

St000812Integer partitions ⟶ ℤ

The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions.

St000813Integer partitions ⟶ ℤ

The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.

St000814Integer partitions ⟶ ℤ

The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions.

St000815Integer partitions ⟶ ℤ

The number of semistandard Young tableaux of partition weight of given shape.

St000835Integer partitions ⟶ ℤ

The minimal difference in size when partitioning the integer partition into two subpartitions.

St000867Integer partitions ⟶ ℤ

The sum of the hook lengths in the first column of an integer partition.

St000869Integer partitions ⟶ ℤ

The sum of the hook lengths of an integer partition.

St000870Integer partitions ⟶ ℤ

The product of the hook lengths of the diagonal cells in an integer partition.

St000897Integer partitions ⟶ ℤ

The number of different multiplicities of parts of an integer partition.

St000901Integer partitions ⟶ ℤ

The cube of the number of standard Young tableaux with shape given by the partition.

St000913Integer partitions ⟶ ℤ

The number of ways to refine the partition into singletons.

St000927Integer partitions ⟶ ℤ

The alternating sum of the coefficients of the character polynomial of an integer partition.

St000928Integer partitions ⟶ ℤ

The sum of the coefficients of the character polynomial of an integer partition.

St000929Integer partitions ⟶ ℤ

The constant term of the character polynomial of an integer partition.

St000933Integer partitions ⟶ ℤ

The number of multipartitions of sizes given by an integer partition.

St000934Integer partitions ⟶ ℤ

The 2-degree of an integer partition.

St000935Integer partitions ⟶ ℤ

The number of ordered refinements of an integer partition.

St000936Integer partitions ⟶ ℤ

The number of even values of the symmetric group character corresponding to the partition.

St000937Integer partitions ⟶ ℤ

The number of positive values of the symmetric group character corresponding to the partition.

St000938Integer partitions ⟶ ℤ

The number of zeros of the symmetric group character corresponding to the partition.

St000939Integer partitions ⟶ ℤ

The number of characters of the symmetric group whose value on the partition is positive.

St000940Integer partitions ⟶ ℤ

The number of characters of the symmetric group whose value on the partition is zero.

St000941Integer partitions ⟶ ℤ

The number of characters of the symmetric group whose value on the partition is even.

St000944Integer partitions ⟶ ℤ

The 3-degree of an integer partition.

St000992Integer partitions ⟶ ℤ

The alternating sum of the parts of an integer partition.

St000993Integer partitions ⟶ ℤ

The multiplicity of the largest part of an integer partition.

St000995Integer partitions ⟶ ℤ

The largest even part of an integer partition.

St000997Integer partitions ⟶ ℤ

The even-odd crank of an integer partition.

St001055Integer partitions ⟶ ℤ

The Grundy value for the game of removing cells of a row in an integer partition.

St001091Integer partitions ⟶ ℤ

The number of parts in an integer partition whose next smaller part has the same size.

St001092Integer partitions ⟶ ℤ

The number of distinct even parts of a partition.

St001097Integer partitions ⟶ ℤ

The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.

St001098Integer partitions ⟶ ℤ

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees.

St001099Integer partitions ⟶ ℤ

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees.

St001100Integer partitions ⟶ ℤ

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees.

St001101Integer partitions ⟶ ℤ

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees.

St001103Integer partitions ⟶ ℤ

The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123.

St001121Integer partitions ⟶ ℤ

The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition.

St001122Integer partitions ⟶ ℤ

The multiplicity of the sign representation in the Kronecker square corresponding to a partition.

St001123Integer partitions ⟶ ℤ

The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.

St001124Integer partitions ⟶ ℤ

The multiplicity of the standard representation in the Kronecker square corresponding to a partition.

St001127Integer partitions ⟶ ℤ

The sum of the squares of the parts of a partition.

St001128Integer partitions ⟶ ℤ

The exponens consonantiae of a partition.

St001129Integer partitions ⟶ ℤ

The product of the squares of the parts of a partition.

St001175Integer partitions ⟶ ℤ

The size of a partition minus the hook length of the base cell.

St001176Integer partitions ⟶ ℤ

The size of a partition minus its first part.

St001177Integer partitions ⟶ ℤ

Twice the mean value of the major index among all standard Young tableaux of a partition.

St001178Integer partitions ⟶ ℤ

Twelve times the variance of the major index among all standard Young tableaux of a partition.

St001214Integer partitions ⟶ ℤ

The aft of an integer partition.

St001247Integer partitions ⟶ ℤ

The number of parts of a partition that are not congruent 2 modulo 3.

St001248Integer partitions ⟶ ℤ

Sum of the even parts of a partition.

St001249Integer partitions ⟶ ℤ

Sum of the odd parts of a partition.

St001250Integer partitions ⟶ ℤ

The number of parts of a partition that are not congruent 0 modulo 3.

St001251Integer partitions ⟶ ℤ

The number of parts of a partition that are not congruent 1 modulo 3.

St001252Integer partitions ⟶ ℤ

Half the sum of the even parts of a partition.

St001262Integer partitions ⟶ ℤ

The dimension of the maximal parabolic seaweed algebra corresponding to the partition.

St001279Integer partitions ⟶ ℤ

The sum of the parts of an integer partition that are at least two.

St001280Integer partitions ⟶ ℤ

The number of parts of an integer partition that are at least two.

St001283Integer partitions ⟶ ℤ

The number of finite solvable groups that are realised by the given partition over the complex numbers.

St001284Integer partitions ⟶ ℤ

The number of finite groups that are realised by the given partition over the complex numbers.

St001360Integer partitions ⟶ ℤ

The number of covering relations in Young's lattice below a partition.

St001364Integer partitions ⟶ ℤ

The number of permutations whose cube equals a fixed permutation of given cycle type.

St001378Integer partitions ⟶ ℤ

The product of the cohook lengths of the integer partition.

St001380Integer partitions ⟶ ℤ

The number of monomer-dimer tilings of a Ferrers diagram.

St001382Integer partitions ⟶ ℤ

The number of boxes in the diagram of a partition that do not lie in its Durfee square.

St001383Integer partitions ⟶ ℤ

The BG-rank of an integer partition.

St001384Integer partitions ⟶ ℤ

The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.

St001385Integer partitions ⟶ ℤ

The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition.

St001387Integer partitions ⟶ ℤ

Number of SYT of the skew shape determined by adding one more box in the first n columns.

St001389Integer partitions ⟶ ℤ

The number of partitions of the same length below the given integer partition.

St001392Integer partitions ⟶ ℤ

The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.

St001400Integer partitions ⟶ ℤ

The total number of Littlewood-Richardson tableaux of given shape.

St001432Integer partitions ⟶ ℤ

The global dimension of the partition.

St001440Integer partitions ⟶ ℤ

The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.

St001442Integer partitions ⟶ ℤ

The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.

St001484Integer partitions ⟶ ℤ

The number of parts that appear precisely once in an integer partition.

St001525Integer partitions ⟶ ℤ

The number of symmetric hooks on the diagonal of a partition.

St001527Integer partitions ⟶ ℤ

The cyclic permutation representation number of an integer partition.

**Ordered trees**(23 statistics) # tree like structures # Catalan objects # graph like objects

St000084Ordered trees ⟶ ℤ

The number of subtrees.

St000085Ordered trees ⟶ ℤ

The number of linear extensions of the tree.

St000094Ordered trees ⟶ ℤ

The depth of an ordered tree.

St000166Ordered trees ⟶ ℤ

The depth minus 1 of an ordered tree.

St000167Ordered trees ⟶ ℤ

The number of leaves of an ordered tree.

St000168Ordered trees ⟶ ℤ

The number of internal nodes of an ordered tree.

St000328Ordered trees ⟶ ℤ

The maximum number of child nodes in a tree.

St000397Ordered trees ⟶ ℤ

The Strahler number of a rooted tree.

St000400Ordered trees ⟶ ℤ

The path length of an ordered tree.

St000410Ordered trees ⟶ ℤ

The tree factorial of an ordered tree.

St000413Ordered trees ⟶ ℤ

The number of ordered trees with the same underlying unordered tree.

St000415Ordered trees ⟶ ℤ

The size of the automorphism group of the rooted tree underlying the ordered tree.

St000416Ordered trees ⟶ ℤ

The number of inequivalent increasing trees of an ordered tree.

St000417Ordered trees ⟶ ℤ

The size of the automorphism group of the ordered tree.

St000521Ordered trees ⟶ ℤ

The number of distinct subtrees of an ordered tree.

St000522Ordered trees ⟶ ℤ

The number of 1-protected nodes of a rooted tree.

St000523Ordered trees ⟶ ℤ

The number of 2-protected nodes of a rooted tree.

St000679Ordered trees ⟶ ℤ

The pruning number of an ordered tree.

St000700Ordered trees ⟶ ℤ

The protection number of an ordered tree.

St000973Ordered trees ⟶ ℤ

The length of the boundary of an ordered tree.

St000974Ordered trees ⟶ ℤ

The length of the trunk of an ordered tree.

St000975Ordered trees ⟶ ℤ

The length of the boundary minus the length of the trunk of an ordered tree.

St001058Ordered trees ⟶ ℤ

The breadth of the ordered tree.

**Parking functions**(10 statistics) # word like objects # path like objects

St000135Parking functions ⟶ ℤ

The number of lucky cars of the parking function.

St000136Parking functions ⟶ ℤ

The dinv of a parking function.

St000165Parking functions ⟶ ℤ

Sum of the entries.

St000188Parking functions ⟶ ℤ

The area of the Dyck path corresponding to a parking function.

St000194Parking functions ⟶ ℤ

The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function.

St000195Parking functions ⟶ ℤ

The number of secondary dinversion pairs of the dyck path corresponding to a parking function.

St000540Parking functions ⟶ ℤ

The sum of the entries of a parking function minus its length.

St000942Parking functions ⟶ ℤ

The number of critical left to right maxima of the parking functions.

St000943Parking functions ⟶ ℤ

The number of spots the most unlucky car had to go further in a parking function.

St001209Parking functions ⟶ ℤ

The pmaj statistic of a parking function.

**Perfect matchings**(39 statistics) # graph like objects

St000041Perfect matchings ⟶ ℤ

The number of nestings of a perfect matching.

St000042Perfect matchings ⟶ ℤ

The number of crossings of a perfect matching.

St000043Perfect matchings ⟶ ℤ

The number of crossings plus two-nestings of a perfect matching.

St000044Perfect matchings ⟶ ℤ

The number of vertices of the unicellular map given by a perfect matching.

St000164Perfect matchings ⟶ ℤ

The number of short pairs.

St000719Perfect matchings ⟶ ℤ

The number of alignments in a perfect matching.

St000720Perfect matchings ⟶ ℤ

The size of the largest partition in the oscillating tableau corresponding to the perfect matching.

St000721Perfect matchings ⟶ ℤ

The sum of the partition sizes in the oscillating tableau corresponding to a perfect matching.

St000746Perfect matchings ⟶ ℤ

The number of pairs with odd minimum in a perfect matching.

St000754Perfect matchings ⟶ ℤ

The Grundy value for the game of removing nestings in a perfect matching.

St000780Perfect matchings ⟶ ℤ

The size of the orbit under rotation of a perfect matching.

St000782Perfect matchings ⟶ ℤ

The indicator function of whether a given perfect matching is an L & P matching.

St000787Perfect matchings ⟶ ℤ

The number of flips required to make a perfect matching noncrossing.

St000788Perfect matchings ⟶ ℤ

The number of nesting-similar perfect matchings of a perfect matching.

St000789Perfect matchings ⟶ ℤ

The number of crossing-similar perfect matchings of a perfect matching.

St000819Perfect matchings ⟶ ℤ

The propagating number of a perfect matching.

St000838Perfect matchings ⟶ ℤ

The number of terminal right-hand endpoints when the vertices are written in order.

St000840Perfect matchings ⟶ ℤ

The number of closers smaller than the largest opener in a perfect matching.

St000841Perfect matchings ⟶ ℤ

The largest opener of a perfect matching.

St000843Perfect matchings ⟶ ℤ

The decomposition number of a perfect matching.

St000924Perfect matchings ⟶ ℤ

The number of topologically connected components of a perfect matching.

St000945Perfect matchings ⟶ ℤ

The number of matchings in the dihedral orbit of a perfect matching.

St001040Perfect matchings ⟶ ℤ

The depth of the decreasing labelled binary unordered tree associated with the perfect matching.

St001041Perfect matchings ⟶ ℤ

The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching.

St001042Perfect matchings ⟶ ℤ

The size of the automorphism group of the leaf labelled binary unordered tree associated with the perfect matching.

St001043Perfect matchings ⟶ ℤ

The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching.

St001044Perfect matchings ⟶ ℤ

The number of pairs whose larger element is at most one more than half the size of the perfect matching.

St001045Perfect matchings ⟶ ℤ

The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching.

St001046Perfect matchings ⟶ ℤ

The maximal number of arcs nesting a given arc of a perfect matching.

St001047Perfect matchings ⟶ ℤ

The maximal number of arcs crossing a given arc of a perfect matching.

St001048Perfect matchings ⟶ ℤ

The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching.

St001049Perfect matchings ⟶ ℤ

The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching.

St001131Perfect matchings ⟶ ℤ

The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching.

St001132Perfect matchings ⟶ ℤ

The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching.

St001133Perfect matchings ⟶ ℤ

The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching.

St001134Perfect matchings ⟶ ℤ

The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching.

St001136Perfect matchings ⟶ ℤ

The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching.

St001152Perfect matchings ⟶ ℤ

The number of pairs with even minimum in a perfect matching.

St001444Perfect matchings ⟶ ℤ

The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching.

**Permutations**(337 statistics) # word like objects

St000001Permutations ⟶ ℤ

The number of reduced words for a permutation.

St000002Permutations ⟶ ℤ

The number of occurrences of the pattern 123 in a permutation.

St000004Permutations ⟶ ℤ

The major index of a permutation.

St000007Permutations ⟶ ℤ

The number of saliances of the permutation.

St000018Permutations ⟶ ℤ

The number of inversions of a permutation.

St000019Permutations ⟶ ℤ

The cardinality of the support of a permutation.

St000020Permutations ⟶ ℤ

The rank of the permutation.

St000021Permutations ⟶ ℤ

The number of descents of a permutation.

St000022Permutations ⟶ ℤ

The number of fixed points of a permutation.

St000023Permutations ⟶ ℤ

The number of inner peaks of a permutation.

St000028Permutations ⟶ ℤ

The number of stack-sorts needed to sort a permutation.

St000029Permutations ⟶ ℤ

The depth of a permutation.

St000030Permutations ⟶ ℤ

The sum of the descent differences of a permutations.

St000031Permutations ⟶ ℤ

The number of cycles in the cycle decomposition of a permutation.

St000033Permutations ⟶ ℤ

The number of permutations greater than or equal to the given permutation in (strong) Bruhat order.

St000034Permutations ⟶ ℤ

The maximum defect over any reduced expression for a permutation and any subexpression.

St000035Permutations ⟶ ℤ

The number of left outer peaks of a permutation.

St000036Permutations ⟶ ℤ

The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.

St000037Permutations ⟶ ℤ

The sign of a permutation.

St000039Permutations ⟶ ℤ

The number of crossings of a permutation.

St000040Permutations ⟶ ℤ

The number of regions of inversion arrangement of a permutation.

St000054Permutations ⟶ ℤ

The first entry of the permutation.

St000055Permutations ⟶ ℤ

The inversion sum of a permutation.

St000056Permutations ⟶ ℤ

The decomposition (or block) number of a permutation.

St000058Permutations ⟶ ℤ

The order of a permutation.

St000060Permutations ⟶ ℤ

The greater neighbor of the maximum.

St000062Permutations ⟶ ℤ

The length of the longest increasing subsequence of the permutation.

St000064Permutations ⟶ ℤ

The number of one-box pattern of a permutation.

St000078Permutations ⟶ ℤ

The number of alternating sign matrices whose left key is the permutation.

St000092Permutations ⟶ ℤ

The number of outer peaks of a permutation.

St000099Permutations ⟶ ℤ

The number of valleys of a permutation, including the boundary.

St000109Permutations ⟶ ℤ

The number of elements less than or equal to the given element in Bruhat order.

St000110Permutations ⟶ ℤ

The number of permutations less than or equal to a permutation in left weak order.

St000111Permutations ⟶ ℤ

The sum of the descent tops (or Genocchi descents) of a permutation.

St000119Permutations ⟶ ℤ

The number of occurrences of the pattern 321 in a permutation.

St000123Permutations ⟶ ℤ

The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.

St000124Permutations ⟶ ℤ

The cardinality of the preimage of the Simion-Schmidt map.

St000133Permutations ⟶ ℤ

The "bounce" of a permutation.

St000141Permutations ⟶ ℤ

The maximum drop size of a permutation.

St000153Permutations ⟶ ℤ

The number of adjacent cycles of a permutation.

St000154Permutations ⟶ ℤ

The sum of the descent bottoms of a permutation.

St000155Permutations ⟶ ℤ

The number of exceedances (also excedences) of a permutation.

St000156Permutations ⟶ ℤ

The Denert index of a permutation.

St000162Permutations ⟶ ℤ

The number of nontrivial cycles in the cycle decomposition of a permutation.

St000209Permutations ⟶ ℤ

Maximum difference of elements in cycles.

St000210Permutations ⟶ ℤ

Minimum over maximum difference of elements in cycles.

St000213Permutations ⟶ ℤ

The number of weak exceedances (also weak excedences) of a permutation.

St000214Permutations ⟶ ℤ

The number of adjacencies of a permutation.

St000215Permutations ⟶ ℤ

The number of adjacencies of a permutation, zero appended.

St000216Permutations ⟶ ℤ

The absolute length of a permutation.

St000217Permutations ⟶ ℤ

The number of occurrences of the pattern 312 in a permutation.

St000218Permutations ⟶ ℤ

The number of occurrences of the pattern 213 in a permutation.

St000219Permutations ⟶ ℤ

The number of occurrences of the pattern 231 in a permutation.

St000220Permutations ⟶ ℤ

The number of occurrences of the pattern 132 in a permutation.

St000221Permutations ⟶ ℤ

The number of strong fixed points of a permutation.

St000222Permutations ⟶ ℤ

The number of alignments in the permutation.

St000223Permutations ⟶ ℤ

The number of nestings in the permutation.

St000224Permutations ⟶ ℤ

The sorting index of a permutation.

St000226Permutations ⟶ ℤ

The convexity of a permutation.

St000234Permutations ⟶ ℤ

The number of global ascents of a permutation.

St000235Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \neq i+1$ considered cyclically.

St000236Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.

St000237Permutations ⟶ ℤ

The number of small exceedances.

St000238Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \notin \{i,i+1\}$.

St000239Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \in \{i,i+1\}$.

St000240Permutations ⟶ ℤ

The number of indices $i$ for which $\pi_i \neq i+1$.

St000241Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i = i+1$ considered cyclically.

St000242Permutations ⟶ ℤ

The number of indices $i$ such that $\pi_i \notin \{ i,i+1 \}$ considered cyclically.

St000243Permutations ⟶ ℤ

The number of cyclic valleys and cyclic peaks of a permutation.

St000245Permutations ⟶ ℤ

The number of ascents of a permutation.

St000246Permutations ⟶ ℤ

The number of non-inversions of a permutation.

St000255Permutations ⟶ ℤ

The number of reduced Kogan faces with the permutation as type.

St000279Permutations ⟶ ℤ

The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations.

St000280Permutations ⟶ ℤ

The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations.

St000304Permutations ⟶ ℤ

The load of a permutation.

St000305Permutations ⟶ ℤ

The inverse major index of a permutation.

St000308Permutations ⟶ ℤ

The height of the tree associated to a permutation.

St000314Permutations ⟶ ℤ

The number of left-to-right-maxima of a permutation.

St000316Permutations ⟶ ℤ

The number of non-left-to-right-maxima of a permutation.

St000317Permutations ⟶ ℤ

The cycle descent number of a permutation.

St000324Permutations ⟶ ℤ

The shape of the tree associated to a permutation.

St000325Permutations ⟶ ℤ

The width of the tree associated to a permutation.

St000333Permutations ⟶ ℤ

The dez statistic, the number of descents of a permutation after replacing fixed points by zeros.

St000334Permutations ⟶ ℤ

The maz index, the major index of a permutation after replacing fixed points by zeros.

St000337Permutations ⟶ ℤ

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.

St000338Permutations ⟶ ℤ

The number of pixed points of a permutation.

St000339Permutations ⟶ ℤ

The maf index of a permutation.

St000341Permutations ⟶ ℤ

The non-inversion sum of a permutation.

St000342Permutations ⟶ ℤ

The cosine of a permutation.

St000352Permutations ⟶ ℤ

The Elizalde-Pak rank of a permutation.

St000353Permutations ⟶ ℤ

The number of inner valleys of a permutation.

St000354Permutations ⟶ ℤ

The number of recoils of a permutation.

St000355Permutations ⟶ ℤ

The number of occurrences of the pattern 21-3.

St000356Permutations ⟶ ℤ

The number of occurrences of the pattern 13-2.

St000357Permutations ⟶ ℤ

The number of occurrences of the pattern 12-3.

St000358Permutations ⟶ ℤ

The number of occurrences of the pattern 31-2.

St000359Permutations ⟶ ℤ

The number of occurrences of the pattern 23-1.

St000360Permutations ⟶ ℤ

The number of occurrences of the pattern 32-1.

St000365Permutations ⟶ ℤ

The number of double ascents of a permutation.

St000366Permutations ⟶ ℤ

The number of double descents of a permutation.

St000367Permutations ⟶ ℤ

The number of simsun double descents of a permutation.

St000371Permutations ⟶ ℤ

The number of mid points of decreasing subsequences of length 3 in a permutation.

St000372Permutations ⟶ ℤ

The number of mid points of increasing subsequences of length 3 in a permutation.

St000373Permutations ⟶ ℤ

The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.

St000374Permutations ⟶ ℤ

The number of exclusive right-to-left minima of a permutation.

St000375Permutations ⟶ ℤ

The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.

St000401Permutations ⟶ ℤ

The size of the symmetry class of a permutation.

St000402Permutations ⟶ ℤ

Half the size of the symmetry class of a permutation.

St000404Permutations ⟶ ℤ

The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation.

St000405Permutations ⟶ ℤ

The number of occurrences of the pattern 1324 in a permutation.

St000406Permutations ⟶ ℤ

The number of occurrences of the pattern 3241 in a permutation.

St000407Permutations ⟶ ℤ

The number of occurrences of the pattern 2143 in a permutation.

St000408Permutations ⟶ ℤ

The number of occurrences of the pattern 4231 in a permutation.

St000423Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 132 in a permutation.

St000424Permutations ⟶ ℤ

The number of occurrences of the pattern 132 or of the pattern 231 in a permutation.

St000425Permutations ⟶ ℤ

The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.

St000426Permutations ⟶ ℤ

The number of occurrences of the pattern 132 or of the pattern 312 in a permutation.

St000427Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 231 in a permutation.

St000428Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 213 in a permutation.

St000429Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 321 in a permutation.

St000430Permutations ⟶ ℤ

The number of occurrences of the pattern 123 or of the pattern 312 in a permutation.

St000431Permutations ⟶ ℤ

The number of occurrences of the pattern 213 or of the pattern 321 in a permutation.

St000432Permutations ⟶ ℤ

The number of occurrences of the pattern 231 or of the pattern 312 in a permutation.

St000433Permutations ⟶ ℤ

The number of occurrences of the pattern 132 or of the pattern 321 in a permutation.

St000434Permutations ⟶ ℤ

The number of occurrences of the pattern 213 or of the pattern 312 in a permutation.

St000435Permutations ⟶ ℤ

The number of occurrences of the pattern 213 or of the pattern 231 in a permutation.

St000436Permutations ⟶ ℤ

The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.

St000437Permutations ⟶ ℤ

The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.

St000440Permutations ⟶ ℤ

The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation.

St000441Permutations ⟶ ℤ

The number of successions of a permutation.

St000446Permutations ⟶ ℤ

The disorder of a permutation.

St000451Permutations ⟶ ℤ

The length of the longest pattern of the form k 1 2.

St000457Permutations ⟶ ℤ

The number of occurrences of one of the patterns 132, 213 or 321 in a permutation.

St000458Permutations ⟶ ℤ

The number of permutations obtained by switching adjacencies or successions.

St000461Permutations ⟶ ℤ

The rix statistic of a permutation.

St000462Permutations ⟶ ℤ

The major index minus the number of excedences of a permutation.

St000463Permutations ⟶ ℤ

The number of admissible inversions of a permutation.

St000470Permutations ⟶ ℤ

The number of runs in a permutation.

St000471Permutations ⟶ ℤ

The sum of the ascent tops of a permutation.

St000472Permutations ⟶ ℤ

The sum of the ascent bottoms of a permutation.

St000483Permutations ⟶ ℤ

The number of times a permutation switches from increasing to decreasing or decreasing to increasing.

St000484Permutations ⟶ ℤ

The sum of St000483 over all subsequences of length at least three.

St000485Permutations ⟶ ℤ

The length of the longest cycle of a permutation.

St000486Permutations ⟶ ℤ

The number of cycles of length at least 3 of a permutation.

St000487Permutations ⟶ ℤ

The length of the shortest cycle of a permutation.

St000488Permutations ⟶ ℤ

The number of cycles of a permutation of length at most 2.

St000489Permutations ⟶ ℤ

The number of cycles of a permutation of length at most 3.

St000494Permutations ⟶ ℤ

The number of inversions of distance at most 3 of a permutation.

St000495Permutations ⟶ ℤ

The number of inversions of distance at most 2 of a permutation.

St000500Permutations ⟶ ℤ

Eigenvalues of the random-to-random operator acting on the regular representation.

St000501Permutations ⟶ ℤ

The size of the first part in the decomposition of a permutation.

St000516Permutations ⟶ ℤ

The number of stretching pairs of a permutation.

St000520Permutations ⟶ ℤ

The number of patterns in a permutation.

St000530Permutations ⟶ ℤ

The number of permutations with the same descent word as the given permutation.

St000534Permutations ⟶ ℤ

The number of 2-rises of a permutation.

St000538Permutations ⟶ ℤ

The number of even inversions of a permutation.

St000539Permutations ⟶ ℤ

The number of odd inversions of a permutation.

St000541Permutations ⟶ ℤ

The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.

St000542Permutations ⟶ ℤ

The number of left-to-right-minima of a permutation.

St000545Permutations ⟶ ℤ

The number of parabolic double cosets with minimal element being the given permutation.

St000546Permutations ⟶ ℤ

The number of global descents of a permutation.

St000570Permutations ⟶ ℤ

The Edelman-Greene number of a permutation.

St000616Permutations ⟶ ℤ

The inversion index of a permutation.

St000619Permutations ⟶ ℤ

The number of cyclic descents of a permutation.

St000622Permutations ⟶ ℤ

The number of occurrences of the patterns 2143 or 4231 in a permutation.

St000623Permutations ⟶ ℤ

The number of occurrences of the pattern 52341 in a permutation.

St000624Permutations ⟶ ℤ

The normalized sum of the minimal distances to a greater element.

St000625Permutations ⟶ ℤ

The sum of the minimal distances to a greater element.

St000638Permutations ⟶ ℤ

The number of up-down runs of a permutation.

St000646Permutations ⟶ ℤ

The number of big ascents of a permutation.

St000647Permutations ⟶ ℤ

The number of big descents of a permutation.

St000648Permutations ⟶ ℤ

The number of 2-excedences of a permutation.

St000649Permutations ⟶ ℤ

The number of 3-excedences of a permutation.

St000650Permutations ⟶ ℤ

The number of 3-rises of a permutation.

St000651Permutations ⟶ ℤ

The maximal size of a rise in a permutation.

St000652Permutations ⟶ ℤ

The maximal difference between successive positions of a permutation.

St000653Permutations ⟶ ℤ

The last descent of a permutation.

St000654Permutations ⟶ ℤ

The first descent of a permutation.

St000662Permutations ⟶ ℤ

The staircase size of the code of a permutation.

St000663Permutations ⟶ ℤ

The number of right floats of a permutation.

St000664Permutations ⟶ ℤ

The number of right ropes of a permutation.

St000665Permutations ⟶ ℤ

The number of rafts of a permutation.

St000666Permutations ⟶ ℤ

The number of right tethers of a permutation.

St000669Permutations ⟶ ℤ

The number of permutations obtained by switching ascents or descents of size 2.

St000670Permutations ⟶ ℤ

The reversal length of a permutation.

St000672Permutations ⟶ ℤ

The number of minimal elements in Bruhat order not less than the permutation.

St000673Permutations ⟶ ℤ

The size of the support of a permutation.

St000677Permutations ⟶ ℤ

The standardized bi-alternating inversion number of a permutation.

St000690Permutations ⟶ ℤ

The size of the conjugacy class of a permutation.

St000692Permutations ⟶ ℤ

Babson and SteingrÃmsson's statistic stat of a permutation.

St000694Permutations ⟶ ℤ

The number of affine bounded permutations that project to a given permutation.

St000696Permutations ⟶ ℤ

The number of cycles in the breakpoint graph of a permutation.

St000702Permutations ⟶ ℤ

The number of weak deficiencies of a permutation.

St000703Permutations ⟶ ℤ

The number of deficiencies of a permutation.

St000709Permutations ⟶ ℤ

The number of occurrences of 14-2-3 or 14-3-2.

St000710Permutations ⟶ ℤ

The number of big deficiencies of a permutation.

St000711Permutations ⟶ ℤ

The number of big exceedences of a permutation.

St000724Permutations ⟶ ℤ

The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.

St000725Permutations ⟶ ℤ

The smallest label of a leaf of the increasing binary tree associated to a permutation.

St000726Permutations ⟶ ℤ

The normalized sum of the leaf labels of the increasing binary tree associated to a permutation.

St000727Permutations ⟶ ℤ

The largest label of a leaf in the binary search tree associated with the permutation.

St000731Permutations ⟶ ℤ

The number of double exceedences of a permutation.

St000732Permutations ⟶ ℤ

The number of double deficiencies of a permutation.

St000740Permutations ⟶ ℤ

The last entry of a permutation.

St000742Permutations ⟶ ℤ

The number of big ascents of a permutation after prepending zero.

St000750Permutations ⟶ ℤ

The number of occurrences of the pattern 4213 in a permutation.

St000751Permutations ⟶ ℤ

The number of occurrences of either of the pattern 2143 or 2143 in a permutation.

St000756Permutations ⟶ ℤ

The sum of the positions of the left to right maxima of a permutation.

St000779Permutations ⟶ ℤ

The tier of a permutation.

St000794Permutations ⟶ ℤ

The mak of a permutation.

St000795Permutations ⟶ ℤ

The mad of a permutation.

St000796Permutations ⟶ ℤ

The stat' of a permutation.

St000797Permutations ⟶ ℤ

The stat`` of a permutation.

St000798Permutations ⟶ ℤ

The makl of a permutation.

St000799Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |213 in a permutation.

St000800Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |231 in a permutation.

St000801Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |312 in a permutation.

St000802Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |321 in a permutation.

St000803Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |132 in a permutation.

St000804Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |123 in a permutation.

St000809Permutations ⟶ ℤ

The reduced reflection length of the permutation.

St000824Permutations ⟶ ℤ

The sum of the number of descents and the number of recoils of a permutation.

St000825Permutations ⟶ ℤ

The sum of the major and the inverse major index of a permutation.

St000828Permutations ⟶ ℤ

The spearman's rho of a permutation and the identity permutation.

St000829Permutations ⟶ ℤ

The Ulam distance of a permutation to the identity permutation.

St000830Permutations ⟶ ℤ

The total displacement of a permutation.

St000831Permutations ⟶ ℤ

The number of indices that are either descents or recoils.

St000832Permutations ⟶ ℤ

The number of permutations obtained by reversing blocks of three consecutive numbers.

St000833Permutations ⟶ ℤ

The comajor index of a permutation.

St000834Permutations ⟶ ℤ

The number of right outer peaks of a permutation.

St000836Permutations ⟶ ℤ

The number of descents of distance 2 of a permutation.

St000837Permutations ⟶ ℤ

The number of ascents of distance 2 of a permutation.

St000842Permutations ⟶ ℤ

The breadth of a permutation.

St000844Permutations ⟶ ℤ

The size of the largest block in the direct sum decomposition of a permutation.

St000862Permutations ⟶ ℤ

The number of parts of the shifted shape of a permutation.

St000863Permutations ⟶ ℤ

The length of the first row of the shifted shape of a permutation.

St000864Permutations ⟶ ℤ

The number of circled entries of the shifted recording tableau of a permutation.

St000866Permutations ⟶ ℤ

The number of admissible inversions of a permutation in the sense of Shareshian-Wachs.

St000868Permutations ⟶ ℤ

The aid statistic in the sense of Shareshian-Wachs.

St000871Permutations ⟶ ℤ

The number of very big ascents of a permutation.

St000872Permutations ⟶ ℤ

The number of very big descents of a permutation.

St000873Permutations ⟶ ℤ

The aix statistic of a permutation.

St000879Permutations ⟶ ℤ

The number of long braid edges in the graph of braid moves of a permutation.

St000880Permutations ⟶ ℤ

The number of connected components of long braid edges in the graph of braid moves of a permutation.

St000881Permutations ⟶ ℤ

The number of short braid edges in the graph of braid moves of a permutation.

St000882Permutations ⟶ ℤ

The number of connected components of short braid edges in the graph of braid moves of a permutation.

St000883Permutations ⟶ ℤ

The number of longest increasing subsequences of a permutation.

St000884Permutations ⟶ ℤ

The number of isolated descents of a permutation.

St000886Permutations ⟶ ℤ

The number of permutations with the same antidiagonal sums.

St000887Permutations ⟶ ℤ

The maximal number of nonzero entries on a diagonal of a permutation matrix.

St000891Permutations ⟶ ℤ

The number of distinct diagonal sums of a permutation matrix.

St000923Permutations ⟶ ℤ

The minimal number with no two order isomorphic substrings of this length in a permutation.

St000956Permutations ⟶ ℤ

The maximal displacement of a permutation.

St000957Permutations ⟶ ℤ

The number of Bruhat lower covers of a permutation.

St000958Permutations ⟶ ℤ

The number of Bruhat factorizations of a permutation.

St000959Permutations ⟶ ℤ

The number of strong Bruhat factorizations of a permutation.

St000961Permutations ⟶ ℤ

The shifted major index of a permutation.

St000962Permutations ⟶ ℤ

The 3-shifted major index of a permutation.

St000963Permutations ⟶ ℤ

The 2-shifted major index of a permutation.

St000988Permutations ⟶ ℤ

The orbit size of a permutation under Foata's bijection.

St000989Permutations ⟶ ℤ

The number of final rises of a permutation.

St000990Permutations ⟶ ℤ

The first ascent of a permutation.

St000991Permutations ⟶ ℤ

The number of right-to-left minima of a permutation.

St000994Permutations ⟶ ℤ

The number of cycle peaks and the number of cycle valleys of a permutation.

St000996Permutations ⟶ ℤ

The number of exclusive left-to-right maxima of a permutation.

St001004Permutations ⟶ ℤ

The number of indices that are either left-to-right maxima or right-to-left minima.

St001005Permutations ⟶ ℤ

The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.

St001052Permutations ⟶ ℤ

The length of the exterior of a permutation.

St001059Permutations ⟶ ℤ

Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation.

St001061Permutations ⟶ ℤ

The number of indices that are both descents and recoils of a permutation.

St001074Permutations ⟶ ℤ

The number of inversions of the cyclic embedding of a permutation.

St001076Permutations ⟶ ℤ

The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12).

St001077Permutations ⟶ ℤ

The prefix exchange distance of a permutation.

St001078Permutations ⟶ ℤ

The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,.

St001079Permutations ⟶ ℤ

The minimal length of a factorization of a permutation using the permutations (12)(34).

St001080Permutations ⟶ ℤ

The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,.

St001081Permutations ⟶ ℤ

The number of minimal length factorizations of a permutation into star transpositions.

St001082Permutations ⟶ ℤ

The number of boxed occurrences of 123 in a permutation.

St001083Permutations ⟶ ℤ

The number of boxed occurrences of 132 in a permutation.

St001084Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |1-23 in a permutation.

St001085Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |21-3 in a permutation.

St001086Permutations ⟶ ℤ

The number of occurrences of the consecutive pattern 132 in a permutation.

St001087Permutations ⟶ ℤ

The number of occurrences of the vincular pattern |12-3 in a permutation.

St001090Permutations ⟶ ℤ

The number of pop-stack-sorts needed to sort a permutation.

St001096Permutations ⟶ ℤ

The size of the overlap set of a permutation.

St001114Permutations ⟶ ℤ

The number of odd descents of a permutation.

St001115Permutations ⟶ ℤ

The number of even descents of a permutation.

St001130Permutations ⟶ ℤ

The number of two successive successions in a permutation.

St001160Permutations ⟶ ℤ

The number of proper blocks (or intervals) of a permutations.

St001162Permutations ⟶ ℤ

The minimum jump of a permutation.

St001168Permutations ⟶ ℤ

The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.

St001171Permutations ⟶ ℤ

The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.

St001174Permutations ⟶ ℤ

The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.

St001207Permutations ⟶ ℤ

The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.

St001208Permutations ⟶ ℤ

The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.

St001220Permutations ⟶ ℤ

The width of a permutation.

St001245Permutations ⟶ ℤ

The cyclic maximal difference between two consecutive entries of a permutation.

St001246Permutations ⟶ ℤ

The maximal difference between two consecutive entries of a permutation.

St001269Permutations ⟶ ℤ

The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.

St001285Permutations ⟶ ℤ

The number of primes in the column sums of the two line notation of a permutation.

St001287Permutations ⟶ ℤ

The number of primes obtained by multiplying preimage and image of a permutation and subtracting one.

St001288Permutations ⟶ ℤ

The number of primes obtained by multiplying preimage and image of a permutation and adding one.

St001293Permutations ⟶ ℤ

The sum of all $1/(i+\pi(i))$ for a permutation $\pi$ times the lcm of all possible values among permutations of the same length.

St001298Permutations ⟶ ℤ

The number of repeated entries in the Lehmer code of a permutation.

St001332Permutations ⟶ ℤ

The number of steps on the non-negative side of the walk associated with the permutation.

St001344Permutations ⟶ ℤ

The neighbouring number of a permutation.

St001346Permutations ⟶ ℤ

The number of parking functions that give the same permutation.

St001359Permutations ⟶ ℤ

The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles.

St001375Permutations ⟶ ℤ

The pancake length of a permutation.

St001377Permutations ⟶ ℤ

The major index minus the number of inversions of a permutation.

St001379Permutations ⟶ ℤ

The number of inversions plus the major index of a permutation.

St001381Permutations ⟶ ℤ

The fertility of a permutation.

St001388Permutations ⟶ ℤ

The number of non-attacking neighbors of a permutation.

St001390Permutations ⟶ ℤ

The number of bumps occurring when Schensted-inserting the letter 1 of a permutation.

St001394Permutations ⟶ ℤ

The genus of a permutation.

St001402Permutations ⟶ ℤ

The number of separators in a permutation.

St001403Permutations ⟶ ℤ

The number of vertical separators in a permutation.

St001405Permutations ⟶ ℤ

The number of bonds in a permutation.

St001411Permutations ⟶ ℤ

The number of patterns 321 or 3412 in a permutation.

St001412Permutations ⟶ ℤ

Number of minimal entries in the Bruhat order matrix of a permutation.

St001439Permutations ⟶ ℤ

The number of even deficiencies and of odd exceedences.

St001461Permutations ⟶ ℤ

The number of topologically connected components of the chord diagram of a permutation.

St001464Permutations ⟶ ℤ

The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.

St001465Permutations ⟶ ℤ

The number of adjacent transpositions in the cycle decomposition of a permutation.

St001466Permutations ⟶ ℤ

The number of transpositions swapping cyclically adjacent numbers in a permutation.

St001468Permutations ⟶ ℤ

The smallest fixpoint of a permutation.

St001469Permutations ⟶ ℤ

The holeyness of a permutation.

St001470Permutations ⟶ ℤ

The cyclic holeyness of a permutation.

St001482Permutations ⟶ ℤ

The product of the prefix sums of a permutation.

St001489Permutations ⟶ ℤ

The maximum of the number of descents and the number of inverse descents.

St001497Permutations ⟶ ℤ

The position of the largest weak excedence of a permutation.

St001511Permutations ⟶ ℤ

The minimal number of transpositions needed to sort a permutation in either direction.

St001513Permutations ⟶ ℤ

The number of nested exceedences of a permutation.

St001516Permutations ⟶ ℤ

The number of cyclic bonds of a permutation.

St001517Permutations ⟶ ℤ

The length of a longest pair of twins in a permutation.

St001519Permutations ⟶ ℤ

The pinnacle sum of a permutation.

St001520Permutations ⟶ ℤ

The number of strict-3-descents.

**Plane partitions**(15 statistics) # partition like objects

St001422Plane partitions ⟶ ℤ

The number of boxes of a plane partition.

St001445Plane partitions ⟶ ℤ

The number of maximal boxes of a plane partition.

St001446Plane partitions ⟶ ℤ

Number of rows in the plane partition.

St001447Plane partitions ⟶ ℤ

Height of the base box of a plane partition.

St001448Plane partitions ⟶ ℤ

Number of odd parts in a plane partition.

St001449Plane partitions ⟶ ℤ

The smallest missing nonzero part in the plane partition.

St001450Plane partitions ⟶ ℤ

The minimum height of a plane partition.

St001451Plane partitions ⟶ ℤ

The side length of the largest cube contained in a plane partition.

St001452Plane partitions ⟶ ℤ

Number of even parts in the plane partition.

St001453Plane partitions ⟶ ℤ

The number of distinct heights in a plane partition.

St001454Plane partitions ⟶ ℤ

The difference between the largest and smallest heights of a plane partition.

St001455Plane partitions ⟶ ℤ

Largest repeated part of a plane partition, and zero if no part is repeated.

St001456Plane partitions ⟶ ℤ

Sum of the top row of a plane partition.

St001457Plane partitions ⟶ ℤ

Multiplicity of the smallest part of a plane partition.

St001460Plane partitions ⟶ ℤ

Number of columns of a plane partition.

**Posets**(61 statistics) # graph like objects

St000068Posets ⟶ ℤ

The number of minimal elements in a poset.

St000069Posets ⟶ ℤ

The number of maximal elements of a poset.

St000070Posets ⟶ ℤ

The number of antichains in a poset.

St000071Posets ⟶ ℤ

The number of maximal chains in a poset.

St000080Posets ⟶ ℤ

The rank of the poset.

St000100Posets ⟶ ℤ

The number of linear extensions of a poset.

St000104Posets ⟶ ℤ

The number of facets in the order polytope of this poset.

St000151Posets ⟶ ℤ

The number of facets in the chain polytope of the poset.

St000180Posets ⟶ ℤ

The number of chains of a poset.

St000181Posets ⟶ ℤ

The number of connected components of the Hasse diagram for the poset.

St000189Posets ⟶ ℤ

The number of elements in the poset.

St000281Posets ⟶ ℤ

The size of the preimage of the map 'to poset' from Binary trees to Posets.

St000282Posets ⟶ ℤ

The size of the preimage of the map 'to poset' from Ordered trees to Posets.

St000298Posets ⟶ ℤ

The order dimension or Dushnik-Miller dimension of a poset.

St000307Posets ⟶ ℤ

The number of rowmotion orbits of a poset.

St000327Posets ⟶ ℤ

The number of cover relations in a poset.

St000524Posets ⟶ ℤ

The number of posets with the same order polynomial.

St000525Posets ⟶ ℤ

The number of posets with the same zeta polynomial.

St000526Posets ⟶ ℤ

The number of posets with combinatorially isomorphic order polytopes.

St000527Posets ⟶ ℤ

The width of the poset.

St000528Posets ⟶ ℤ

The height of a poset.

St000550Posets ⟶ ℤ

The number of modular elements of a lattice.

St000551Posets ⟶ ℤ

The number of left modular elements of a lattice.

St000632Posets ⟶ ℤ

The jump number of the poset.

St000633Posets ⟶ ℤ

The size of the automorphism group of a poset.

St000634Posets ⟶ ℤ

The number of endomorphisms of a poset.

St000635Posets ⟶ ℤ

The number of strictly order preserving maps of a poset into itself.

St000639Posets ⟶ ℤ

The number of relations in a poset.

St000640Posets ⟶ ℤ

The rank of the largest boolean interval in a poset.

St000641Posets ⟶ ℤ

The number of non-empty boolean intervals in a poset.

St000642Posets ⟶ ℤ

The size of the smallest orbit of antichains under Panyushev complementation.

St000643Posets ⟶ ℤ

The size of the largest orbit of antichains under Panyushev complementation.

St000656Posets ⟶ ℤ

The number of cuts of a poset.

St000680Posets ⟶ ℤ

The Grundy value for Hackendot on posets.

St000717Posets ⟶ ℤ

The number of ordinal summands of a poset.

St000845Posets ⟶ ℤ

The maximal number of elements covered by an element in a poset.

St000846Posets ⟶ ℤ

The maximal number of elements covering an element of a poset.

St000848Posets ⟶ ℤ

The balance constant multiplied with the number of linear extensions of a poset.

St000849Posets ⟶ ℤ

The number of 1/3-balanced pairs in a poset.

St000850Posets ⟶ ℤ

The number of 1/2-balanced pairs in a poset.

St000906Posets ⟶ ℤ

The length of the shortest maximal chain in a poset.

St000907Posets ⟶ ℤ

The number of maximal antichains of minimal length in a poset.

St000908Posets ⟶ ℤ

The length of the shortest maximal antichain in a poset.

St000909Posets ⟶ ℤ

The number of maximal chains of maximal size in a poset.

St000910Posets ⟶ ℤ

The number of maximal chains of minimal length in a poset.

St000911Posets ⟶ ℤ

The number of maximal antichains of maximal size in a poset.

St000912Posets ⟶ ℤ

The number of maximal antichains in a poset.

St000914Posets ⟶ ℤ

The sum of the values of the MÃ¶bius function of a poset.

St001095Posets ⟶ ℤ

The number of non-isomorphic posets with precisely one further covering relation.

St001105Posets ⟶ ℤ

The number of greedy linear extensions of a poset.

St001106Posets ⟶ ℤ

The number of supergreedy linear extensions of a poset.

St001268Posets ⟶ ℤ

The size of the largest ordinal summand in the poset.

St001300Posets ⟶ ℤ

The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.

St001301Posets ⟶ ℤ

The first Betti number of the order complex associated with the poset.

St001343Posets ⟶ ℤ

The dimension of the reduced incidence algebra of a poset.

St001396Posets ⟶ ℤ

Number of triples of incomparable elements in a finite poset.

St001397Posets ⟶ ℤ

Number of pairs of incomparable elements in a finite poset.

St001398Posets ⟶ ℤ

Number of subsets of size 3 of elements in a poset that form a "v".

St001399Posets ⟶ ℤ

The distinguishing number of a poset.

St001472Posets ⟶ ℤ

The permanent of the Coxeter matrix of the poset.

St001510Posets ⟶ ℤ

The number of self-evacuating linear extensions of a finite poset.

**Semistandard tableaux**(16 statistics) # tableau like objects

St000101Semistandard tableaux ⟶ ℤ

The cocharge of a semistandard tableau.

St000102Semistandard tableaux ⟶ ℤ

The charge of a semistandard tableau.

St000103Semistandard tableaux ⟶ ℤ

The sum of the entries of a semistandard tableau.

St000112Semistandard tableaux ⟶ ℤ

The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$.

St000116Semistandard tableaux ⟶ ℤ

The major index of a semistandard tableau obtained by standardizing.

St000170Semistandard tableaux ⟶ ℤ

The trace of a semistandard tableau.

St000173Semistandard tableaux ⟶ ℤ

The segment statistic of a semistandard tableau.

St000174Semistandard tableaux ⟶ ℤ

The flush statistic of a semistandard tableau.

St000736Semistandard tableaux ⟶ ℤ

The last entry in the first row of a semistandard tableau.

St000737Semistandard tableaux ⟶ ℤ

The last entry on the main diagonal of a semistandard tableau.

St000739Semistandard tableaux ⟶ ℤ

The first entry in the last row of a semistandard tableau.

St001401Semistandard tableaux ⟶ ℤ

The number of distinct entries in a semistandard tableau.

St001407Semistandard tableaux ⟶ ℤ

The number of minimal entries in a semistandard tableau.

St001408Semistandard tableaux ⟶ ℤ

The number of maximal entries in a semistandard tableau.

St001409Semistandard tableaux ⟶ ℤ

The maximal entry of a semistandard tableau.

St001410Semistandard tableaux ⟶ ℤ

The minimal entry of a semistandard tableau.

**Set partitions**(101 statistics) # partition like objects

St000105Set partitions ⟶ ℤ

The number of blocks in the set partition.

St000163Set partitions ⟶ ℤ

The size of the orbit of the set partition under rotation.

St000211Set partitions ⟶ ℤ

The rank of the set partition.

St000229Set partitions ⟶ ℤ

Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition.

St000230Set partitions ⟶ ℤ

Sum of the minimal elements of the blocks of a set partition.

St000231Set partitions ⟶ ℤ

Sum of the maximal elements of the blocks of a set partition.

St000232Set partitions ⟶ ℤ

The number of crossings of a set partition.

St000233Set partitions ⟶ ℤ

The number of nestings of a set partition.

St000247Set partitions ⟶ ℤ

The number of singleton blocks of a set partition.

St000248Set partitions ⟶ ℤ

The number of anti-singletons of a set partition.

St000249Set partitions ⟶ ℤ

The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition.

St000250Set partitions ⟶ ℤ

The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition.

St000251Set partitions ⟶ ℤ

The number of nonsingleton blocks of a set partition.

St000253Set partitions ⟶ ℤ

The crossing number of a set partition.

St000254Set partitions ⟶ ℤ

The nesting number of a set partition.

St000490Set partitions ⟶ ℤ

The intertwining number of a set partition.

St000491Set partitions ⟶ ℤ

The number of inversions of a set partition.

St000492Set partitions ⟶ ℤ

The rob statistic of a set partition.

St000493Set partitions ⟶ ℤ

The los statistic of a set partition.

St000496Set partitions ⟶ ℤ

The rcs statistic of a set partition.

St000497Set partitions ⟶ ℤ

The lcb statistic of a set partition.

St000498Set partitions ⟶ ℤ

The lcs statistic of a set partition.

St000499Set partitions ⟶ ℤ

The rcb statistic of a set partition.

St000502Set partitions ⟶ ℤ

The number of successions of a set partitions.

St000503Set partitions ⟶ ℤ

The maximal difference between two elements in a common block.

St000504Set partitions ⟶ ℤ

The cardinality of the first block of a set partition.

St000505Set partitions ⟶ ℤ

The biggest entry in the block containing the 1.

St000554Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,2},{3}} in a set partition.

St000555Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} in a set partition.

St000556Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} in a set partition.

St000557Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} in a set partition.

St000558Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,2}} in a set partition.

St000559Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.

St000560Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.

St000561Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,2,3}} in a set partition.

St000562Set partitions ⟶ ℤ

The number of internal points of a set partition.

St000563Set partitions ⟶ ℤ

The number of overlapping pairs of blocks of a set partition.

St000564Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} in a set partition.

St000565Set partitions ⟶ ℤ

The major index of a set partition.

St000572Set partitions ⟶ ℤ

The dimension exponent of a set partition.

St000573Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element.

St000574Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element.

St000575Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton.

St000576Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element.

St000577Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.

St000578Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton.

St000579Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.

St000580Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal.

St000581Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal.

St000582Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block.

St000583Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal.

St000584Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal.

St000585Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block.

St000586Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.

St000587Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal.

St000588Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal.

St000589Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.

St000590Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block.

St000591Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal.

St000592Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal.

St000593Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal.

St000594Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block.

St000595Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal.

St000596Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal.

St000597Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block.

St000598Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block.

St000599Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block.

St000600Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block.

St000601Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block.

St000602Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal.

St000603Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal.

St000604Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal.

St000605Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block.

St000606Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block.

St000607Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block.

St000608Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal.

St000609Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

St000610Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal.

St000611Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal.

St000612Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.

St000613Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block.

St000614Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block.

St000615Set partitions ⟶ ℤ

The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal.

St000695Set partitions ⟶ ℤ

The number of blocks in the first part of the atomic decomposition of a set partition.

St000728Set partitions ⟶ ℤ

The dimension of a set partition.

St000729Set partitions ⟶ ℤ

The minimal arc length of a set partition.

St000730Set partitions ⟶ ℤ

The maximal arc length of a set partition.

St000747Set partitions ⟶ ℤ

A variant of the major index of a set partition.

St000748Set partitions ⟶ ℤ

The major index of the permutation obtained by flattening the set partition.

St000793Set partitions ⟶ ℤ

The length of the longest partition in the vacillating tableau corresponding to a set partition.

St000823Set partitions ⟶ ℤ

The number of unsplittable factors of the set partition.

St000839Set partitions ⟶ ℤ

The largest opener of a set partition.

St000925Set partitions ⟶ ℤ

The number of topologically connected components of a set partition.

St000971Set partitions ⟶ ℤ

The smallest closer of a set partition.

St001050Set partitions ⟶ ℤ

The number of terminal closers of a set partition.

St001051Set partitions ⟶ ℤ

The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition.

St001062Set partitions ⟶ ℤ

The maximal size of a block of a set partition.

St001075Set partitions ⟶ ℤ

The minimal size of a block of a set partition.

St001094Set partitions ⟶ ℤ

The depth index of a set partition.

St001151Set partitions ⟶ ℤ

The number of blocks with odd minimum.

St001153Set partitions ⟶ ℤ

The number of blocks with even minimum in a set partition.

**Signed permutations**(6 statistics) # word like objects

St001427Signed permutations ⟶ ℤ

The number of descents of a signed permutation.

St001428Signed permutations ⟶ ℤ

The number of B-inversions of a signed permutation.

St001429Signed permutations ⟶ ℤ

The number of negative entries in a signed permutation.

St001430Signed permutations ⟶ ℤ

The number of positive entries in a signed permutation.

St001433Signed permutations ⟶ ℤ

The flag major index of a signed permutation.

St001434Signed permutations ⟶ ℤ

The number of negative sum pairs of a signed permutation.

**Skew partitions**(5 statistics) # partition like objects

St001435Skew partitions ⟶ ℤ

The number of missing boxes in the first row.

St001438Skew partitions ⟶ ℤ

The number of missing boxes of a skew partition.

St001487Skew partitions ⟶ ℤ

The number of inner corners of a skew partition.

St001488Skew partitions ⟶ ℤ

The number of corners of a skew partition.

St001490Skew partitions ⟶ ℤ

The number of connected components of a skew partition.

**Standard tableaux**(21 statistics) # tableau like objects

St000009Standard tableaux ⟶ ℤ

The charge of a standard tableau.

St000016Standard tableaux ⟶ ℤ

The number of attacking pairs of a standard tableau.

St000017Standard tableaux ⟶ ℤ

The number of inversions of a standard tableau.

St000057Standard tableaux ⟶ ℤ

The Shynar inversion number of a standard tableau.

St000059Standard tableaux ⟶ ℤ

The inversion number of a standard tableau as defined by Haglund and Stevens.

St000075Standard tableaux ⟶ ℤ

The orbit size of a standard tableau under promotion.

St000157Standard tableaux ⟶ ℤ

The number of descents of a standard tableau.

St000169Standard tableaux ⟶ ℤ

The cocharge of a standard tableau.

St000330Standard tableaux ⟶ ℤ

The (standard) major index of a standard tableau.

St000336Standard tableaux ⟶ ℤ

The leg major index of a standard tableau.

St000507Standard tableaux ⟶ ℤ

The number of ascents of a standard tableau.

St000508Standard tableaux ⟶ ℤ

Eigenvalues of the random-to-random operator acting on a simple module.

St000693Standard tableaux ⟶ ℤ

The modular (standard) major index of a standard tableau.

St000733Standard tableaux ⟶ ℤ

The row containing the largest entry of a standard tableau.

St000734Standard tableaux ⟶ ℤ

The last entry in the first row of a standard tableau.

St000735Standard tableaux ⟶ ℤ

The last entry on the main diagonal of a standard tableau.

St000738Standard tableaux ⟶ ℤ

The first entry in the last row of a standard tableau.

St000743Standard tableaux ⟶ ℤ

The number of entries in a standard Young tableau such that the next integer is a neighbour.

St000744Standard tableaux ⟶ ℤ

The length of the path to the largest entry in a standard Young tableau.

St000745Standard tableaux ⟶ ℤ

The index of the last row whose first entry is the row number in a standard Young tableau.

St001462Standard tableaux ⟶ ℤ

The number of factors of a standard tableaux under concatenation.