Identifier
There are 1210 statistics in the database, and possibly some more waiting for verification:
   Alternating sign matrices (22 statistics) # matrix like objects
St000065 Alternating sign matrices ⟶ ℤ
The number of entries equal to -1 in an alternating sign matrix.
St000066 Alternating sign matrices ⟶ ℤ
The column of the unique '1' in the first row of the alternating sign matrix.
St000067 Alternating sign matrices ⟶ ℤ
The inversion number of the alternating sign matrix.
St000076 Alternating sign matrices ⟶ ℤ
The rank of the alternating sign matrix in the alternating sign matrix poset.
St000134 Alternating sign matrices ⟶ ℤ
The size of the orbit of an alternating sign matrix under gyration.
St000187 Alternating sign matrices ⟶ ℤ
The determinant of an alternating sign matrix.
St000193 Alternating sign matrices ⟶ ℤ
The row of the unique '1' in the first column of the alternating sign matrix.
St000197 Alternating sign matrices ⟶ ℤ
The number of entries equal to positive one in the alternating sign matrix.
St000199 Alternating sign matrices ⟶ ℤ
The column of the unique '1' in the last row of the alternating sign matrix.
St000200 Alternating sign matrices ⟶ ℤ
The row of the unique '1' in the last column of the alternating sign matrix.
St000227 Alternating sign matrices ⟶ ℤ
The osculating paths major index of an alternating sign matrix.
St000332 Alternating sign matrices ⟶ ℤ
The positive inversions of an alternating sign matrix.
St000888 Alternating sign matrices ⟶ ℤ
The maximal sum of entries on a diagonal of an alternating sign matrix.
St000889 Alternating sign matrices ⟶ ℤ
The number of alternating sign matrices with the same antidiagonal sums.
St000890 Alternating sign matrices ⟶ ℤ
The number of nonzero entries in an alternating sign matrix.
St000892 Alternating sign matrices ⟶ ℤ
The maximal number of nonzero entries on a diagonal of an alternating sign matrix....
St000893 Alternating sign matrices ⟶ ℤ
The number of distinct diagonal sums of an alternating sign matrix.
St000894 Alternating sign matrices ⟶ ℤ
The trace of an alternating sign matrix.
St000895 Alternating sign matrices ⟶ ℤ
The number of ones on the main diagonal of an alternating sign matrix.
St000896 Alternating sign matrices ⟶ ℤ
The number of zeros on the main diagonal of an alternating sign matrix.
St000898 Alternating sign matrices ⟶ ℤ
The number of maximal entries in the last diagonal of the monotone triangle.
St001030 Alternating sign matrices ⟶ ℤ
Half the number of non-boundary horizontal edges in the fully packed loop corresp....
   Binary trees (36 statistics) # tree like structures # Catalan objects # graph like objects
St000045 Binary trees ⟶ ℤ
The number of linear extensions of a binary tree.
St000050 Binary trees ⟶ ℤ
The depth or height of a binary tree.
St000051 Binary trees ⟶ ℤ
The size of the left subtree of a binary tree.
St000061 Binary trees ⟶ ℤ
The number of nodes on the left branch of a binary tree.
St000082 Binary trees ⟶ ℤ
The number of elements smaller than a binary tree in Tamari order.
St000083 Binary trees ⟶ ℤ
The number of left oriented leafs of a binary tree except the first one.
St000118 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary ....
St000121 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a bin....
St000122 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a bin....
St000125 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[[[.,.],.],.
St000126 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a....
St000127 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a....
St000128 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a....
St000129 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a....
St000130 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a....
St000131 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],.
St000132 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a....
St000161 Binary trees ⟶ ℤ
The sum of the sizes of the right subtrees of a binary tree.
St000196 Binary trees ⟶ ℤ
The number of occurrences of the contiguous pattern [[.,.],[.,.
St000198 Binary trees ⟶ ℤ
A decimal representation of a binary tree as a code word.
St000201 Binary trees ⟶ ℤ
The number of leaf nodes in a binary tree.
St000203 Binary trees ⟶ ℤ
The number of external nodes of a binary tree.
St000204 Binary trees ⟶ ℤ
The number of internal nodes of a binary tree.
St000252 Binary trees ⟶ ℤ
The number of nodes of degree 3 of a binary tree.
St000385 Binary trees ⟶ ℤ
The number of vertices with out-degree 1 in a binary tree.
St000396 Binary trees ⟶ ℤ
The register function (or Horton-Strahler number) of a binary tree.
St000398 Binary trees ⟶ ℤ
The sum of the depths of the vertices (or total internal path length) of a binary....
St000399 Binary trees ⟶ ℤ
The external path length of a binary tree.
St000409 Binary trees ⟶ ℤ
The number of pitchforks in a binary tree.
St000411 Binary trees ⟶ ℤ
The tree factorial of a binary tree.
St000412 Binary trees ⟶ ℤ
The number of binary trees with the same underlying unordered tree.
St000414 Binary trees ⟶ ℤ
The binary logarithm of the number of binary trees with the same underlying unord....
St000568 Binary trees ⟶ ℤ
The hook number of a binary tree.
St000569 Binary trees ⟶ ℤ
The sum of the heights of the vertices of a binary tree.
St000701 Binary trees ⟶ ℤ
The protection number of a binary tree.
St000919 Binary trees ⟶ ℤ
The number of maximal left branches of a binary tree.
   Binary words (44 statistics) # word like objects # path like objects
St000288 Binary words ⟶ ℤ
The number of ones in a binary word.
St000289 Binary words ⟶ ℤ
The decimal representation of a binary word.
St000290 Binary words ⟶ ℤ
The major index of a binary word.
St000291 Binary words ⟶ ℤ
The number of descents of a binary word.
St000292 Binary words ⟶ ℤ
The number of ascents of a binary word.
St000293 Binary words ⟶ ℤ
The number of inversions of a binary word.
St000294 Binary words ⟶ ℤ
The number of distinct factors of a binary word.
St000295 Binary words ⟶ ℤ
The length of the border of a binary word.
St000296 Binary words ⟶ ℤ
The length of the symmetric border of a binary word.
St000297 Binary words ⟶ ℤ
The number of leading ones in a binary word.
St000326 Binary words ⟶ ℤ
The position of the first one in a binary word after appending a 1 at the end.
St000347 Binary words ⟶ ℤ
The inversion sum of a binary word.
St000348 Binary words ⟶ ℤ
The non-inversion sum of a binary word.
St000389 Binary words ⟶ ℤ
The number of runs of ones of odd length in a binary word.
St000390 Binary words ⟶ ℤ
The number of runs of ones in a binary word.
St000391 Binary words ⟶ ℤ
The sum of the positions of the ones in a binary word.
St000392 Binary words ⟶ ℤ
The length of the longest run of ones in a binary word.
St000393 Binary words ⟶ ℤ
The number of strictly increasing runs in a binary word.
St000518 Binary words ⟶ ℤ
The number of distinct subsequences in a binary word.
St000519 Binary words ⟶ ℤ
The largest length of a factor maximising the subword complexity.
St000529 Binary words ⟶ ℤ
The number of permutations whose descent word is the given binary word.
St000543 Binary words ⟶ ℤ
The size of the conjugacy class of a binary word.
St000626 Binary words ⟶ ℤ
The minimal period of a binary word.
St000627 Binary words ⟶ ℤ
The exponent of a binary word.
St000628 Binary words ⟶ ℤ
The balance of a binary word.
St000629 Binary words ⟶ ℤ
The defect of a binary word.
St000630 Binary words ⟶ ℤ
The length of the shortest palindromic decomposition of a binary word.
St000631 Binary words ⟶ ℤ
The number of distinct palindromic decompositions of a binary word.
St000682 Binary words ⟶ ℤ
The Grundy value of Welter's game on a binary word.
St000691 Binary words ⟶ ℤ
The number of changes of a binary word.
St000753 Binary words ⟶ ℤ
The Grundy value for the game of Kayles on a binary word.
St000792 Binary words ⟶ ℤ
The Grundy value for the game of ruler on a binary word.
St000826 Binary words ⟶ ℤ
The stopping time of the decimal representation of the binary word for the 3x+1 p....
St000827 Binary words ⟶ ℤ
The decimal representation of a binary word with a leading 1.
St000847 Binary words ⟶ ℤ
The number of standard Young tableaux whose descent set is the binary word.
St000875 Binary words ⟶ ℤ
The semilength of the longest Dyck word in the Catalan factorisation of a binary ....
St000876 Binary words ⟶ ℤ
The number of factors in the Catalan decomposition of a binary word.
St000877 Binary words ⟶ ℤ
The depth of the binary word interpreted as a path.
St000878 Binary words ⟶ ℤ
The number of ones minus the number of zeros of a binary word.
St000885 Binary words ⟶ ℤ
The number of critical steps in the Catalan decomposition of a binary word.
St000921 Binary words ⟶ ℤ
The number of internal inversions of a binary word.
St000922 Binary words ⟶ ℤ
The minimal number such that all substrings of this length are unique.
St000982 Binary words ⟶ ℤ
The length of the longest constant subword.
St000983 Binary words ⟶ ℤ
The length of the longest alternating subword.
   Cores (5 statistics) # partition like objects
St000158 Cores ⟶ ℤ
The length of a core.
St000190 Cores ⟶ ℤ
The size of a core.
St000191 Cores ⟶ ℤ
The number of strong covers of a core.
St000192 Cores ⟶ ℤ
Number of covers of a core in weak Bruhat order.
St000202 Cores ⟶ ℤ
The number of k-cores contained in the k-core.
   Dyck paths (185 statistics) # Catalan objects # path like objects
St000005 Dyck paths ⟶ ℤ
The bounce statistic of a Dyck path.
St000006 Dyck paths ⟶ ℤ
The dinv statistic of a Dyck path.
St000011 Dyck paths ⟶ ℤ
The number of touch points of a Dyck path.
St000012 Dyck paths ⟶ ℤ
The area of a Dyck path.
St000013 Dyck paths ⟶ ℤ
The height of a Dyck path.
St000014 Dyck paths ⟶ ℤ
The number of parking functions supported by a Dyck path.
St000015 Dyck paths ⟶ ℤ
The number of peaks of a Dyck path.
St000024 Dyck paths ⟶ ℤ
The number of double up and double down steps of a Dyck path.
St000025 Dyck paths ⟶ ℤ
The number of initial rises of a Dyck path.
St000026 Dyck paths ⟶ ℤ
The position of the first return of a Dyck path.
St000027 Dyck paths ⟶ ℤ
The major index of a Dyck path.
St000032 Dyck paths ⟶ ℤ
The number of elements smaller than the given Dyck path in the Tamari Order.
St000038 Dyck paths ⟶ ℤ
The product of the heights of the descending steps of a Dyck path.
St000052 Dyck paths ⟶ ℤ
The number of valleys of a Dyck path not on the x-axis.
St000053 Dyck paths ⟶ ℤ
The number of valleys of the Dyck path.
St000079 Dyck paths ⟶ ℤ
The number of alternating sign matrices for a given Dyck path.
St000117 Dyck paths ⟶ ℤ
The number of centered tunnels of a Dyck path.
St000120 Dyck paths ⟶ ℤ
The number of left tunnels of a Dyck path.
St000144 Dyck paths ⟶ ℤ
The pyramid weight of the Dyck path.
St000306 Dyck paths ⟶ ℤ
The bounce count of a Dyck path.
St000329 Dyck paths ⟶ ℤ
The number of evenly positioned ascents of the Dyck path, with the initial positi....
St000331 Dyck paths ⟶ ℤ
The number of upper interactions of a Dyck path.
St000335 Dyck paths ⟶ ℤ
The difference of lower and upper interactions.
St000340 Dyck paths ⟶ ℤ
The number of non-final maximal sub-paths of length greater than one.
St000369 Dyck paths ⟶ ℤ
The dinv deficit of a Dyck path.
St000376 Dyck paths ⟶ ℤ
The bounce deficit of a Dyck path.
St000386 Dyck paths ⟶ ℤ
The number of factors DDU in a Dyck path.
St000394 Dyck paths ⟶ ℤ
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
St000395 Dyck paths ⟶ ℤ
The sum of the heights of the peaks of a Dyck path.
St000418 Dyck paths ⟶ ℤ
The number of Dyck paths that are weakly below a Dyck path.
St000419 Dyck paths ⟶ ℤ
The number of Dyck paths that are weakly above the Dyck path, except for the path....
St000420 Dyck paths ⟶ ℤ
The number of Dyck paths that are weakly above a Dyck path.
St000421 Dyck paths ⟶ ℤ
The number of Dyck paths that are weakly below a Dyck path, except for the path i....
St000438 Dyck paths ⟶ ℤ
The position of the last up step in a Dyck path.
St000439 Dyck paths ⟶ ℤ
The position of the first down step of a Dyck path.
St000442 Dyck paths ⟶ ℤ
The maximal area to the right of an up step of a Dyck path.
St000443 Dyck paths ⟶ ℤ
The number of long tunnels of a Dyck path.
St000444 Dyck paths ⟶ ℤ
The length of the maximal rise of a Dyck path.
St000445 Dyck paths ⟶ ℤ
The number of rises of length 1 of a Dyck path.
St000476 Dyck paths ⟶ ℤ
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
St000617 Dyck paths ⟶ ℤ
The number of global maxima of a Dyck path.
St000645 Dyck paths ⟶ ℤ
The sum of the areas of the rectangles formed by two consecutive peaks and the va....
St000655 Dyck paths ⟶ ℤ
The length of the minimal rise of a Dyck path.
St000658 Dyck paths ⟶ ℤ
The number of rises of length 2 of a Dyck path.
St000659 Dyck paths ⟶ ℤ
The number of rises of length at least 2 of a Dyck path.
St000660 Dyck paths ⟶ ℤ
The number of rises of length at least 3 of a Dyck path.
St000661 Dyck paths ⟶ ℤ
The number of rises of length 3 of a Dyck path.
St000674 Dyck paths ⟶ ℤ
The number of hills of a Dyck path.
St000675 Dyck paths ⟶ ℤ
The number of centered multitunnels of a Dyck path.
St000676 Dyck paths ⟶ ℤ
The number of odd rises of a Dyck path.
St000678 Dyck paths ⟶ ℤ
The number of up steps after the last double rise of a Dyck path.
St000683 Dyck paths ⟶ ℤ
The number of points below the Dyck path such that the diagonal to the north-east....
St000684 Dyck paths ⟶ ℤ
The global dimension of the LNakayama algebra associated to a Dyck path.
St000685 Dyck paths ⟶ ℤ
The dominant dimension of the LNakayama algebra associated to a Dyck path.
St000686 Dyck paths ⟶ ℤ
The finitistic dominant dimension of a Dyck path.
St000687 Dyck paths ⟶ ℤ
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.
St000688 Dyck paths ⟶ ℤ
The global dimension minus the dominant dimension of the LNakayama algebra associ....
St000689 Dyck paths ⟶ ℤ
The maximal n such that the minimal generator-cogenerator module in the LNakayama....
St000790 Dyck paths ⟶ ℤ
The number of pairs of centered tunnels, one strictly containing the other, of a ....
St000791 Dyck paths ⟶ ℤ
The number of pairs of left tunnels, one strictly containing the other, of a Dyck....
St000874 Dyck paths ⟶ ℤ
The position of the last double rise in a Dyck path.
St000920 Dyck paths ⟶ ℤ
The logarithmic height of a Dyck path.
St000930 Dyck paths ⟶ ℤ
The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver.....
St000931 Dyck paths ⟶ ℤ
The number of occurrences of the pattern UUU in a Dyck path.
St000932 Dyck paths ⟶ ℤ
The number of occurrences of the pattern UDU in a Dyck path.
St000946 Dyck paths ⟶ ℤ
The sum of the skew hook positions in a Dyck path.
St000947 Dyck paths ⟶ ℤ
The major index east count of a Dyck path.
St000949 Dyck paths ⟶ ℤ
Gives the number of generalised tilting modules of the corresponding LNakayama al....
St000950 Dyck paths ⟶ ℤ
Number of tilting modules of the corresponding LNakayama algebra, where a tilting....
St000951 Dyck paths ⟶ ℤ
The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra.
St000952 Dyck paths ⟶ ℤ
Gives the number of irreducible factors of the Coxeter polynomial of the Dyck pat....
St000953 Dyck paths ⟶ ℤ
The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck....
St000954 Dyck paths ⟶ ℤ
Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$.
St000955 Dyck paths ⟶ ℤ
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama a....
St000964 Dyck paths ⟶ ℤ
Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when....
St000965 Dyck paths ⟶ ℤ
The sum of the dimension of Ext^i(D(A),A) for i=1,.
St000966 Dyck paths ⟶ ℤ
Number of peaks minus the global dimension of the corresponding LNakayama algebra....
St000967 Dyck paths ⟶ ℤ
The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra....
St000968 Dyck paths ⟶ ℤ
We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dy....
St000969 Dyck paths ⟶ ℤ
We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dy....
St000970 Dyck paths ⟶ ℤ
Number of peaks minus the dominant dimension of the corresponding LNakayama algeb....
St000976 Dyck paths ⟶ ℤ
The sum of the positions of double up-steps of a Dyck path.
St000977 Dyck paths ⟶ ℤ
MacMahon's equal index of a Dyck path.
St000978 Dyck paths ⟶ ℤ
The sum of the positions of double down-steps of a Dyck path.
St000979 Dyck paths ⟶ ℤ
Half of MacMahon's equal index of a Dyck path.
St000980 Dyck paths ⟶ ℤ
The number of boxes weakly below the path and above the diagonal that lie below a....
St000981 Dyck paths ⟶ ℤ
The length of the longest zigzag subpath.
St000984 Dyck paths ⟶ ℤ
The number of boxes below precisely one peak.
St000998 Dyck paths ⟶ ℤ
Number of indecomposable projective modules with injective dimension smaller than....
St000999 Dyck paths ⟶ ℤ
Number of indecomposable projective module with injective dimension equal to the ....
St001000 Dyck paths ⟶ ℤ
Number of indecomposable modules with projective dimension equal to the global di....
St001001 Dyck paths ⟶ ℤ
The number of indecomposable modules with projective and injective dimension equa....
St001002 Dyck paths ⟶ ℤ
Number of indecomposable modules with projective and injective dimension at most ....
St001003 Dyck paths ⟶ ℤ
The number of indecomposable modules with projective dimension at most 1 in the N....
St001006 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension equal to the global dimension ....
St001007 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension 1 in the Nakayama algebra corr....
St001008 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with projective dimension 1 in the Nak....
St001009 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with projective dimension g when g is ....
St001010 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with projective dimension g-1 when g i....
St001011 Dyck paths ⟶ ℤ
Number of simple modules of projective dimension 2 in the Nakayama algebra corres....
St001012 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension at most 2 in the Nakayama alge....
St001013 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with codominant dimension equal to the....
St001014 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with codominant dimension equal to the....
St001015 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with codominant dimension equal to one....
St001016 Dyck paths ⟶ ℤ
Number of indecomosable injective modules with codominant dimension at most 1 in ....
St001017 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with projective dimension equal to the....
St001018 Dyck paths ⟶ ℤ
Sum of projective dimension of the indecomposable injective modules of the Nakaya....
St001019 Dyck paths ⟶ ℤ
Sum of the projective dimensions of the simple modules in the Nakayama algebra co....
St001020 Dyck paths ⟶ ℤ
Sum of the codominant dimensions of the non-projective indecomposable injective m....
St001021 Dyck paths ⟶ ℤ
Sum of the differences between projective and codominant dimension of the non-pro....
St001022 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension 3 in the Nakayama algebra corr....
St001023 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension at most 3 in the Nakayama alge....
St001024 Dyck paths ⟶ ℤ
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corr....
St001025 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension 4 in the Nakayama algebra corr....
St001026 Dyck paths ⟶ ℤ
The maximum of the projective dimensions of the indecomposable non-projective inj....
St001027 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension equal to injective dimension i....
St001028 Dyck paths ⟶ ℤ
Number of simple modules with injective dimension equal to the dominant dimension....
St001031 Dyck paths ⟶ ℤ
The height of the bicoloured Motzkin path associated with the Dyck path.
St001032 Dyck paths ⟶ ℤ
The number of horizontal steps in the bicoloured Motzkin path associated with the....
St001033 Dyck paths ⟶ ℤ
The normalized area of the parallelogram polyomino associated with the Dyck path.....
St001034 Dyck paths ⟶ ℤ
The area of the parallelogram polyomino associated with the Dyck path.
St001035 Dyck paths ⟶ ℤ
The convexity degree of the parallelogram polyomino associated with the Dyck path....
St001036 Dyck paths ⟶ ℤ
The number of inner corners of the parallelogram polyomino associated with the Dy....
St001037 Dyck paths ⟶ ℤ
The number of inner corners of the upper path of the parallelogram polyomino asso....
St001038 Dyck paths ⟶ ℤ
The minimal height of a column in the parallelogram polyomino associated with the....
St001039 Dyck paths ⟶ ℤ
The maximal height of a column in the parallelogram polyomino associated with a D....
St001063 Dyck paths ⟶ ℤ
Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.
St001064 Dyck paths ⟶ ℤ
Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy ....
St001065 Dyck paths ⟶ ℤ
Number of indecomposable reflexive modules in the corresponding Nakayama algebra.....
St001066 Dyck paths ⟶ ℤ
The number of simple reflexive modules in the corresponding Nakayama algebra.
St001067 Dyck paths ⟶ ℤ
The number of simple modules of dominant dimension at least two in the correspond....
St001068 Dyck paths ⟶ ℤ
Number of torsionless simple modules in the corresponding Nakayama algebra.
St001088 Dyck paths ⟶ ℤ
Number of indecomposable projective non-injective modules with dominant dimension....
St001089 Dyck paths ⟶ ℤ
Number of indecomposable projective non-injective modules minus the number of ind....
St001104 Dyck paths ⟶ ℤ
The number of descents of the invariant in a tensor power of the adjoint represen....
St001107 Dyck paths ⟶ ℤ
The number of times one can erase the first up and the last down step in a Dyck p....
St001113 Dyck paths ⟶ ℤ
Number of indecomposable projective non-injective modules with reflexive Auslande....
St001125 Dyck paths ⟶ ℤ
The number of simple modules that satisfy the 2-regular condition in the correspo....
St001126 Dyck paths ⟶ ℤ
Number of simple module that are 1-regular in the corresponding Nakayama algebra.....
St001135 Dyck paths ⟶ ℤ
The projective dimension of the first simple module in the Nakayama algebra corre....
St001137 Dyck paths ⟶ ℤ
Number of simple modules that are 3-regular in the corresponding Nakayama algebra....
St001138 Dyck paths ⟶ ℤ
The number of indecomposable modules with projective dimension or injective dimen....
St001139 Dyck paths ⟶ ℤ
The number of occurrences of hills of size 2 in a Dyck path.
St001140 Dyck paths ⟶ ℤ
Number of indecomposable modules with projective and injective dimension at least....
St001141 Dyck paths ⟶ ℤ
The number of occurrences of hills of size 3 in a Dyck path.
St001142 Dyck paths ⟶ ℤ
The projective dimension of the socle of the regular module as a bimodule in the ....
St001159 Dyck paths ⟶ ℤ
Number of simple modules with dominant dimension equal to the global dimension in....
St001161 Dyck paths ⟶ ℤ
The major index north count of a Dyck path.
St001163 Dyck paths ⟶ ℤ
The number of simple modules with dominant dimension at least three in the corres....
St001164 Dyck paths ⟶ ℤ
Number of indecomposable injective modules whose socle has projective dimension a....
St001165 Dyck paths ⟶ ℤ
Number of simple modules with even projective dimension in the corresponding Naka....
St001166 Dyck paths ⟶ ℤ
Number of indecomposable projective non-injective modules with dominant dimension....
St001167 Dyck paths ⟶ ℤ
The number of simple modules with projective dimension at least 3 in the correspo....
St001169 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension at least two in the correspond....
St001170 Dyck paths ⟶ ℤ
Number of indecomposable injective modules whose socle has projective dimension a....
St001172 Dyck paths ⟶ ℤ
The number of 1-rises at odd height of a Dyck path.
St001179 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with projective dimension at most 2 in....
St001180 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with projective dimension at most 1.
St001181 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with grade at least 3 in the correspon....
St001182 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with codominant dimension at least two....
St001183 Dyck paths ⟶ ℤ
The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algeb....
St001184 Dyck paths ⟶ ℤ
Number of indecomposable injective modules with grade at least 1 in the correspon....
St001185 Dyck paths ⟶ ℤ
The number of indecomposable injective modules of grade at least 2 in the corresp....
St001186 Dyck paths ⟶ ℤ
Number of simple modules with grade at least 3 in the corresponding Nakayama alge....
St001187 Dyck paths ⟶ ℤ
The number of simple modules with grade at least one in the corresponding Nakayam....
St001188 Dyck paths ⟶ ℤ
The number of simple modules $S$ with grade \....
St001189 Dyck paths ⟶ ℤ
The number of simple modules with dominant and codominant dimension equal to zero....
St001190 Dyck paths ⟶ ℤ
Number of simple modules with projective dimension at most 4 in the corresponding....
St001191 Dyck paths ⟶ ℤ
Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corre....
St001192 Dyck paths ⟶ ℤ
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the correspondin....
St001193 Dyck paths ⟶ ℤ
The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such th....
St001194 Dyck paths ⟶ ℤ
The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ ....
St001195 Dyck paths ⟶ ℤ
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra A....
St001196 Dyck paths ⟶ ℤ
The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding....
St001197 Dyck paths ⟶ ℤ
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal....
St001198 Dyck paths ⟶ ℤ
The number of simple modules in the algebra $eAe$ with projective dimension at most....
St001199 Dyck paths ⟶ ℤ
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minim....
St001200 Dyck paths ⟶ ℤ
The number of simple modules in $eAe$ with projective dimension at most 2 in the co....
St001201 Dyck paths ⟶ ℤ
The grade of the simple module $S_0$ in the special CNakayama algebra corresponding....
St001202 Dyck paths ⟶ ℤ
Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch s....
St001203 Dyck paths ⟶ ℤ
We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) wit....
St001204 Dyck paths ⟶ ℤ
Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch s....
St001205 Dyck paths ⟶ ℤ
The number of "peaks of the peaks".
St001206 Dyck paths ⟶ ℤ
The maximal dimension of an indecomposable projective $eAe$-module (that is the hei....
St001210 Dyck paths ⟶ ℤ
Gives the maximal vector space dimension of the first Ext-group between an indeco....
   Finite Cartan types (36 statistics) # Cartan type objects
St000106 Finite Cartan types ⟶ ℤ
The size of the associated Weyl group.
St000107 Finite Cartan types ⟶ ℤ
The dimension of the representation $V(\Lambda_1)$.
St000113 Finite Cartan types ⟶ ℤ
The rank of the Cartan type.
St000138 Finite Cartan types ⟶ ℤ
The Catalan number of an irreducible finite Cartan type.
St000139 Finite Cartan types ⟶ ℤ
The Coxeter number of a finite Cartan type.
St000140 Finite Cartan types ⟶ ℤ
The positive Catalan number of an irreducible finite Cartan type.
St000821 Finite Cartan types ⟶ ℤ
The determinant of the Cartan matrix.
St000851 Finite Cartan types ⟶ ℤ
The third Fuss-Catalan number of a finite Cartan type.
St000852 Finite Cartan types ⟶ ℤ
The second Fuss-Catalan number of a finite Cartan type.
St000853 Finite Cartan types ⟶ ℤ
The number of almost positive roots of a finite Cartan type.
St000854 Finite Cartan types ⟶ ℤ
The number of orbits of reflections of a finite Cartan type.
St000855 Finite Cartan types ⟶ ℤ
The number of full-support reflections in the Weyl group of a finite Cartan type.....
St000856 Finite Cartan types ⟶ ℤ
The number of conjugacy classes in the Weyl group of a finite Cartan type.
St000857 Finite Cartan types ⟶ ℤ
The number of reflections of the Weyl group of a finite Cartan type.
St000858 Finite Cartan types ⟶ ℤ
The number of factorizations of any Coxeter element into reflections of a finite ....
St000859 Finite Cartan types ⟶ ℤ
The number of parking functions of a finite Cartan type.
St000860 Finite Cartan types ⟶ ℤ
The size of the center of the Weyl group of a finite Cartan type.
St000861 Finite Cartan types ⟶ ℤ
The maximal dimension of an irreducible representation of the Weyl group of a fin....
St000865 Finite Cartan types ⟶ ℤ
The number of Coxeter elements in the Weyl group of a finite Cartan type.
St000960 Finite Cartan types ⟶ ℤ
The permanent of the Cartan matrix of a finite Cartan type.
St001053 Finite Cartan types ⟶ ℤ
The second positive Fuss-Catalan number of a finite Cartan type.
St001054 Finite Cartan types ⟶ ℤ
The third positive Fuss-Catalan number of a finite Cartan type.
St001143 Finite Cartan types ⟶ ℤ
The number of pairs in the Weyl group of given type with mu-coefficient of the Ka....
St001144 Finite Cartan types ⟶ ℤ
The largest mu-coefficient of the Kazhdan Lusztig polynomial occurring in the Wey....
St001145 Finite Cartan types ⟶ ℤ
The largest coefficient in a Kazhdan Lusztig polynomial of the Weyl group of give....
St001146 Finite Cartan types ⟶ ℤ
The number of Grassmannian elements in the Coxeter group of the given type.
St001147 Finite Cartan types ⟶ ℤ
The number of minuscule dominant weights in the weight lattice of a finite Cartan....
St001148 Finite Cartan types ⟶ ℤ
The dimension of the adjoint representation of the Lie group of given type.
St001149 Finite Cartan types ⟶ ℤ
The dimension of the quasi-minuscule representation of the Lie group of given typ....
St001150 Finite Cartan types ⟶ ℤ
The minimal dimension of a faithful linear representation of the Lie algebra of g....
St001154 Finite Cartan types ⟶ ℤ
The dual Coxeter number of a finite Cartan type.
St001155 Finite Cartan types ⟶ ℤ
The number of conjugacy classes of subgroups of the Weyl group of given type.
St001156 Finite Cartan types ⟶ ℤ
The Dynkin index of the Lie algebra of given type.
St001157 Finite Cartan types ⟶ ℤ
The exponent of the Weyl group of given type.
St001158 Finite Cartan types ⟶ ℤ
The size of the mutation class of quivers of given type.
St001173 Finite Cartan types ⟶ ℤ
The number of commutative positive roots in the root system of the given finite C....
   Gelfand-Tsetlin patterns (11 statistics) # matrix like objects
St000072 Gelfand-Tsetlin patterns ⟶ ℤ
The number of circled entries.
St000073 Gelfand-Tsetlin patterns ⟶ ℤ
The number of boxed entries.
St000074 Gelfand-Tsetlin patterns ⟶ ℤ
The number of special entries.
St000077 Gelfand-Tsetlin patterns ⟶ ℤ
The number of boxed and circled entries.
St000114 Gelfand-Tsetlin patterns ⟶ ℤ
The sum of the entries of the Gelfand-Tsetlin pattern.
St000115 Gelfand-Tsetlin patterns ⟶ ℤ
The single entry in the last row.
St000152 Gelfand-Tsetlin patterns ⟶ ℤ
The number of boxed plus the number of special entries.
St000176 Gelfand-Tsetlin patterns ⟶ ℤ
The total number of tiles in the Gelfand-Tsetlin pattern.
St000177 Gelfand-Tsetlin patterns ⟶ ℤ
The number of free tiles in the pattern.
St000178 Gelfand-Tsetlin patterns ⟶ ℤ
Number of free entries.
St000186 Gelfand-Tsetlin patterns ⟶ ℤ
The sum of the first row in a Gelfand-Tsetlin pattern.
   Graphs (134 statistics) # graph like objects
St000081 Graphs ⟶ ℤ
The number of edges of a graph.
St000086 Graphs ⟶ ℤ
The number of subgraphs.
St000087 Graphs ⟶ ℤ
The number of induced subgraphs.
St000093 Graphs ⟶ ℤ
The length of the maximal independent set of vertices of a graph.
St000095 Graphs ⟶ ℤ
The number of triangles of a graph.
St000096 Graphs ⟶ ℤ
The number of spanning trees of a graph.
St000097 Graphs ⟶ ℤ
The order of the largest clique of the graph.
St000098 Graphs ⟶ ℤ
The chromatic number of a graph.
St000171 Graphs ⟶ ℤ
The degree of the graph.
St000172 Graphs ⟶ ℤ
The Grundy number of a graph.
St000244 Graphs ⟶ ℤ
The cardinality of the automorphism group of a graph.
St000258 Graphs ⟶ ℤ
The burning number of a graph.
St000259 Graphs ⟶ ℤ
The diameter of a connected graph.
St000260 Graphs ⟶ ℤ
The radius of a connected graph.
St000261 Graphs ⟶ ℤ
The edge connectivity of a graph.
St000262 Graphs ⟶ ℤ
The vertex connectivity of a graph.
St000263 Graphs ⟶ ℤ
The Szeged index of a graph.
St000264 Graphs ⟶ ℤ
The girth of a graph, which is not a tree.
St000265 Graphs ⟶ ℤ
The Wiener index of a graph.
St000266 Graphs ⟶ ℤ
The number of spanning subgraphs of a graph with the same connected components.
St000267 Graphs ⟶ ℤ
The number of maximal spanning forests contained in a graph.
St000268 Graphs ⟶ ℤ
The number of strongly connected orientations of a graph.
St000269 Graphs ⟶ ℤ
The number of acyclic orientations of a graph.
St000270 Graphs ⟶ ℤ
The number of forests contained in a graph.
St000271 Graphs ⟶ ℤ
The chromatic index of a connected graph.
St000272 Graphs ⟶ ℤ
The treewidth of a graph.
St000273 Graphs ⟶ ℤ
The domination number of a graph.
St000274 Graphs ⟶ ℤ
The number of perfect matchings of a graph.
St000276 Graphs ⟶ ℤ
The size of the preimage of the map 'to graph' from Ordered trees to Graphs.
St000283 Graphs ⟶ ℤ
The size of the preimage of the map 'to graph' from Binary trees to Graphs.
St000286 Graphs ⟶ ℤ
The number of connected components of the complement of a graph.
St000287 Graphs ⟶ ℤ
The number of connected components of a graph.
St000299 Graphs ⟶ ℤ
The number of nonisomorphic vertex-induced subtrees.
St000300 Graphs ⟶ ℤ
The number of independent sets of vertices of a graph.
St000301 Graphs ⟶ ℤ
The number of facets of the stable set polytope of a graph.
St000302 Graphs ⟶ ℤ
The determinant of the distance matrix of a connected graph.
St000303 Graphs ⟶ ℤ
The determinant of the product of the incidence matrix and its transpose of a gra....
St000309 Graphs ⟶ ℤ
The number of vertices with even degree.
St000310 Graphs ⟶ ℤ
The minimal degree of a vertex of a graph.
St000311 Graphs ⟶ ℤ
The number of vertices of odd degree in a graph.
St000312 Graphs ⟶ ℤ
The number of leaves in a graph.
St000313 Graphs ⟶ ℤ
The number of degree 2 vertices of a graph.
St000315 Graphs ⟶ ℤ
The number of isolated vertices of a graph.
St000322 Graphs ⟶ ℤ
The skewness of a graph.
St000323 Graphs ⟶ ℤ
The minimal crossing number of a graph.
St000343 Graphs ⟶ ℤ
The number of spanning subgraphs of a graph.
St000344 Graphs ⟶ ℤ
The number of strongly connected outdegree sequences of a graph.
St000349 Graphs ⟶ ℤ
The number of different adjacency matrices of a graph.
St000350 Graphs ⟶ ℤ
The sum of the vertex degrees of a graph.
St000351 Graphs ⟶ ℤ
The determinant of the adjacency matrix of a graph.
St000361 Graphs ⟶ ℤ
The second Zagreb index of a graph.
St000362 Graphs ⟶ ℤ
The size of a minimal vertex cover of a graph.
St000363 Graphs ⟶ ℤ
The number of minimal vertex covers of a graph.
St000364 Graphs ⟶ ℤ
The exponent of the automorphism group of a graph.
St000368 Graphs ⟶ ℤ
The Altshuler-Steinberg determinant of a graph.
St000370 Graphs ⟶ ℤ
The genus of a graph.
St000379 Graphs ⟶ ℤ
The number of Hamiltonian cycles in a graph.
St000387 Graphs ⟶ ℤ
The matching number of a graph.
St000388 Graphs ⟶ ℤ
The number of orbits of vertices of a graph under automorphisms.
St000403 Graphs ⟶ ℤ
The Szeged index minus the Wiener index of a graph.
St000422 Graphs ⟶ ℤ
The energy of a graph, if it is integral.
St000447 Graphs ⟶ ℤ
The number of pairs of vertices of a graph with distance 3.
St000448 Graphs ⟶ ℤ
The number of pairs of vertices of a graph with distance 2.
St000449 Graphs ⟶ ℤ
The number of pairs of vertices of a graph with distance 4.
St000450 Graphs ⟶ ℤ
The number of edges minus the number of vertices plus 2 of a graph.
St000452 Graphs ⟶ ℤ
The number of distinct eigenvalues of a graph.
St000453 Graphs ⟶ ℤ
The number of distinct Laplacian eigenvalues of a graph.
St000454 Graphs ⟶ ℤ
The largest eigenvalue of a graph if it is integral.
St000455 Graphs ⟶ ℤ
The second largest eigenvalue of a graph if it is integral.
St000456 Graphs ⟶ ℤ
The monochromatic index of a connected graph.
St000464 Graphs ⟶ ℤ
The Schultz index of a connected graph.
St000465 Graphs ⟶ ℤ
The first Zagreb index of a graph.
St000466 Graphs ⟶ ℤ
The Gutman (or modified Schultz) index of a connected graph.
St000467 Graphs ⟶ ℤ
The hyper-Wiener index of a connected graph.
St000468 Graphs ⟶ ℤ
The Hosoya index of a graph.
St000469 Graphs ⟶ ℤ
The distinguishing number of a graph.
St000479 Graphs ⟶ ℤ
The Ramsey number of a graph.
St000482 Graphs ⟶ ℤ
The (zero)-forcing number of a graph.
St000535 Graphs ⟶ ℤ
The rank-width of a graph.
St000536 Graphs ⟶ ℤ
The pathwidth of a graph.
St000537 Graphs ⟶ ℤ
The cutwidth of a graph.
St000544 Graphs ⟶ ℤ
The cop number of a graph.
St000552 Graphs ⟶ ℤ
The number of cut vertices of a graph.
St000553 Graphs ⟶ ℤ
The number of blocks of a connected graph.
St000571 Graphs ⟶ ℤ
The F-index (or forgotten topological index) of a graph.
St000636 Graphs ⟶ ℤ
The hull number of a graph.
St000637 Graphs ⟶ ℤ
The length of the longest cycle in a graph.
St000671 Graphs ⟶ ℤ
The maximin edge-connectivity for choosing a subgraph.
St000699 Graphs ⟶ ℤ
The toughness times the least common multiple of 1,.
St000718 Graphs ⟶ ℤ
The largest Laplacian eigenvalue of a graph if it is integral.
St000722 Graphs ⟶ ℤ
The number of different neighbourhoods in a graph.
St000723 Graphs ⟶ ℤ
The maximal cardinality of a set of vertices with the same neighbourhood in a gra....
St000741 Graphs ⟶ ℤ
The Colin de Verdière graph invariant.
St000771 Graphs ⟶ ℤ
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.....
St000772 Graphs ⟶ ℤ
The multiplicity of the largest distance Laplacian eigenvalue in a connected grap....
St000773 Graphs ⟶ ℤ
The multiplicity of the largest Laplacian eigenvalue in a graph.
St000774 Graphs ⟶ ℤ
The maximal multiplicity of a Laplacian eigenvalue in a graph.
St000775 Graphs ⟶ ℤ
The multiplicity of the largest eigenvalue in a graph.
St000776 Graphs ⟶ ℤ
The maximal multiplicity of an eigenvalue in a graph.
St000777 Graphs ⟶ ℤ
The number of distinct eigenvalues of the distance Laplacian of a connected graph....
St000778 Graphs ⟶ ℤ
The metric dimension of a graph.
St000785 Graphs ⟶ ℤ
The number of distinct colouring schemes of a graph.
St000786 Graphs ⟶ ℤ
The maximal number of occurrences of a colour in a proper colouring of a graph.
St000822 Graphs ⟶ ℤ
The Hadwiger number of the graph.
St000915 Graphs ⟶ ℤ
The Ore degree of a graph.
St000916 Graphs ⟶ ℤ
The packing number of a graph.
St000917 Graphs ⟶ ℤ
The open packing number of a graph.
St000918 Graphs ⟶ ℤ
The 2-limited packing number of a graph.
St000926 Graphs ⟶ ℤ
The clique-coclique number of a graph.
St000948 Graphs ⟶ ℤ
The chromatic discriminant of a graph.
St000972 Graphs ⟶ ℤ
The composition number of a graph.
St000985 Graphs ⟶ ℤ
The number of positive eigenvalues of the adjacency matrix of the graph.
St000986 Graphs ⟶ ℤ
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
St000987 Graphs ⟶ ℤ
The number of positive eigenvalues of the Laplacian matrix of the graph.
St001029 Graphs ⟶ ℤ
The size of the core of a graph.
St001056 Graphs ⟶ ℤ
The Grundy value for the game of deleting vertices of a graph until it has no edg....
St001057 Graphs ⟶ ℤ
The Grundy value of the game of creating an independent set in a graph.
St001060 Graphs ⟶ ℤ
The distinguishing index of a graph.
St001069 Graphs ⟶ ℤ
The coefficient of the monomial xy of the Tutte polynomial of the graph.
St001070 Graphs ⟶ ℤ
The absolute value of the derivative of the chromatic polynomial of the graph at ....
St001071 Graphs ⟶ ℤ
The beta invariant of the graph.
St001072 Graphs ⟶ ℤ
The evaluation of the Tutte polynomial of the graph at x and y equal to 3.
St001073 Graphs ⟶ ℤ
The absolute value of the evaluation of the Tutte polynomial of the graph at x eq....
St001093 Graphs ⟶ ℤ
The detour number of a graph.
St001108 Graphs ⟶ ℤ
The 2-dynamic chromatic number of a graph.
St001109 Graphs ⟶ ℤ
Then number of proper colourings of a graph with as few colours as possible.
St001110 Graphs ⟶ ℤ
The 3-dynamic chromatic number of a graph.
St001111 Graphs ⟶ ℤ
The weak 2-dynamic chromatic number of a graph.
St001112 Graphs ⟶ ℤ
The 3-weak dynamic number of a graph.
St001116 Graphs ⟶ ℤ
The game chromatic number of a graph.
St001117 Graphs ⟶ ℤ
The game chromatic index of a graph.
St001118 Graphs ⟶ ℤ
The acyclic chromatic index of a graph.
St001119 Graphs ⟶ ℤ
The length of a shortest maximal path in a graph.
St001120 Graphs ⟶ ℤ
The length of a longest path in a graph.
   Integer compositions (38 statistics) # partition like objects
St000008 Integer compositions ⟶ ℤ
The major index of the composition.
St000047 Integer compositions ⟶ ℤ
The number of standard immaculate tableaux of a given shape.
St000089 Integer compositions ⟶ ℤ
The absolute variation of a composition.
St000090 Integer compositions ⟶ ℤ
The variation of a composition.
St000091 Integer compositions ⟶ ℤ
The descent variation of a composition.
St000277 Integer compositions ⟶ ℤ
The size of the preimage of the map 'descent composition' Mp00071descent composition from Permutatio....
St000285 Integer compositions ⟶ ℤ
The size of the preimage of the map 'to inverse des composition' from Parking fun....
St000381 Integer compositions ⟶ ℤ
The largest part of an integer composition.
St000382 Integer compositions ⟶ ℤ
The first part of an integer composition.
St000383 Integer compositions ⟶ ℤ
The last part of an integer composition.
St000657 Integer compositions ⟶ ℤ
The smallest part of an integer composition.
St000757 Integer compositions ⟶ ℤ
The length of the longest weakly inreasing subsequence of parts of an integer com....
St000758 Integer compositions ⟶ ℤ
The length of the longest staircase fitting into an integer composition.
St000760 Integer compositions ⟶ ℤ
The length of the longest strictly decreasing subsequence of parts of an integer ....
St000761 Integer compositions ⟶ ℤ
The number of ascents in an integer composition.
St000762 Integer compositions ⟶ ℤ
The sum of the positions of the weak records of an integer composition.
St000763 Integer compositions ⟶ ℤ
The sum of the positions of the strong records of an integer composition.
St000764 Integer compositions ⟶ ℤ
The number of strong records in an integer composition.
St000765 Integer compositions ⟶ ℤ
The number of weak records in an integer composition.
St000766 Integer compositions ⟶ ℤ
The number of inversions of an integer composition.
St000767 Integer compositions ⟶ ℤ
The number of runs in an integer composition.
St000768 Integer compositions ⟶ ℤ
The number of peaks in an integer composition.
St000769 Integer compositions ⟶ ℤ
The major index of a composition.
St000805 Integer compositions ⟶ ℤ
The number of peaks of the associated bargraph.
St000806 Integer compositions ⟶ ℤ
The semiperimeter of the associated bargraph.
St000807 Integer compositions ⟶ ℤ
The sum of the heights of the valleys of the associated bargraph.
St000808 Integer compositions ⟶ ℤ
The number of up steps of the associated bargraph.
St000816 Integer compositions ⟶ ℤ
The number of standard composition tableaux of the composition.
St000817 Integer compositions ⟶ ℤ
The sum of the entries in the column specified by the composition of the change o....
St000818 Integer compositions ⟶ ℤ
The sum of the entries in the column specified by the composition of the change o....
St000820 Integer compositions ⟶ ℤ
The number of compositions obtained by rotating the composition.
St000899 Integer compositions ⟶ ℤ
The maximal number of repetitions of an integer composition.
St000900 Integer compositions ⟶ ℤ
The minimal number of repetitions of a part in an integer composition.
St000902 Integer compositions ⟶ ℤ
The minimal number of repetitions of an integer composition.
St000903 Integer compositions ⟶ ℤ
The number of different parts of an integer composition.
St000904 Integer compositions ⟶ ℤ
The maximal number of repetitions of an integer composition.
St000905 Integer compositions ⟶ ℤ
The number of different multiplicities of parts of an integer composition.
St001102 Integer compositions ⟶ ℤ
The number of words with multiplicities of the letters given by the composition, ....
   Integer partitions (150 statistics) # partition like objects
St000003 Integer partitions ⟶ ℤ
The number of standard Young tableaux of the partition.
St000010 Integer partitions ⟶ ℤ
The length of the partition.
St000046 Integer partitions ⟶ ℤ
The largest eigenvalue of the random to random operator acting on the simple modu....
St000048 Integer partitions ⟶ ℤ
The multinomial of the parts of a partition.
St000049 Integer partitions ⟶ ℤ
The number of set partitions whose sorted block sizes correspond to the partition....
St000063 Integer partitions ⟶ ℤ
The number of linear extensions of a certain poset defined from a partition \la....
St000088 Integer partitions ⟶ ℤ
The row sums of the character table of the symmetric group.
St000108 Integer partitions ⟶ ℤ
The number of partitions contained in the given partition.
St000137 Integer partitions ⟶ ℤ
The Grundy value of an integer partition.
St000142 Integer partitions ⟶ ℤ
The number of even parts of a partition.
St000143 Integer partitions ⟶ ℤ
The largest repeated part of a partition.
St000145 Integer partitions ⟶ ℤ
The Dyson rank of a partition.
St000146 Integer partitions ⟶ ℤ
The Andrews-Garvan crank of a partition.
St000147 Integer partitions ⟶ ℤ
The largest part of an integer partition.
St000148 Integer partitions ⟶ ℤ
The number of odd parts of a partition.
St000149 Integer partitions ⟶ ℤ
The number of cells of the partition whose leg is zero and arm is odd.
St000150 Integer partitions ⟶ ℤ
The floored half-sum of the multiplicities of a partition.
St000159 Integer partitions ⟶ ℤ
The number of distinct parts of the integer partition.
St000160 Integer partitions ⟶ ℤ
Multiplicity of the smallest part of $\lambda$.
St000175 Integer partitions ⟶ ℤ
Degree of the polynomial counting the number of semistandard Young tableaux when ....
St000179 Integer partitions ⟶ ℤ
The product of the hook lengths of the integer partition.
St000182 Integer partitions ⟶ ℤ
The number of permutations whose cycle type is the given integer partition.
St000183 Integer partitions ⟶ ℤ
The side length of the Durfee square of an integer partition.
St000184 Integer partitions ⟶ ℤ
The size of the centralizer of any permutation of given cycle type.
St000185 Integer partitions ⟶ ℤ
The weighted size of a partition.
St000205 Integer partitions ⟶ ℤ
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and part....
St000206 Integer partitions ⟶ ℤ
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and inte....
St000207 Integer partitions ⟶ ℤ
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....
St000208 Integer partitions ⟶ ℤ
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....
St000212 Integer partitions ⟶ ℤ
The number of standard Young tableaux for an integer partition such that no two c....
St000225 Integer partitions ⟶ ℤ
Difference between largest and smallest parts in a partition.
St000228 Integer partitions ⟶ ℤ
The size of a partition.
St000256 Integer partitions ⟶ ℤ
The number of parts from which one can substract 2 and still get an integer parti....
St000257 Integer partitions ⟶ ℤ
The number of distinct parts of a partition that occur at least twice.
St000275 Integer partitions ⟶ ℤ
Number of permutations whose sorted list of non zero multiplicities of the Lehmer....
St000278 Integer partitions ⟶ ℤ
The size of the preimage of the map 'to partition' from Integer compositions to I....
St000284 Integer partitions ⟶ ℤ
The Plancherel distribution on integer partitions.
St000318 Integer partitions ⟶ ℤ
The number of addable cells of the Ferrers diagram of an integer partition.
St000319 Integer partitions ⟶ ℤ
The spin of an integer partition.
St000320 Integer partitions ⟶ ℤ
The dinv adjustment of an integer partition.
St000321 Integer partitions ⟶ ℤ
The number of integer partitions of n that are dominated by an integer partition.....
St000345 Integer partitions ⟶ ℤ
The number of refinements of a partition.
St000346 Integer partitions ⟶ ℤ
The number of coarsenings of a partition.
St000377 Integer partitions ⟶ ℤ
The dinv defect of an integer partition.
St000378 Integer partitions ⟶ ℤ
The diagonal inversion number of an integer partition.
St000380 Integer partitions ⟶ ℤ
Half the perimeter of the largest rectangle that fits inside the diagram of an in....
St000384 Integer partitions ⟶ ℤ
The maximal part of the shifted composition of an integer partition.
St000459 Integer partitions ⟶ ℤ
The hook length of the base cell of a partition.
St000460 Integer partitions ⟶ ℤ
The hook length of the last cell along the main diagonal of an integer partition.....
St000473 Integer partitions ⟶ ℤ
The number of parts of a partition that are strictly bigger than the number of on....
St000474 Integer partitions ⟶ ℤ
Dyson's crank of a partition.
St000475 Integer partitions ⟶ ℤ
The number of parts equal to 1 in a partition.
St000477 Integer partitions ⟶ ℤ
The weight of a partition according to Alladi.
St000478 Integer partitions ⟶ ℤ
Another weight of a partition according to Alladi.
St000480 Integer partitions ⟶ ℤ
The number of lower covers of a partition in dominance order.
St000481 Integer partitions ⟶ ℤ
The number of upper covers of a partition in dominance order.
St000506 Integer partitions ⟶ ℤ
The number of standard desarrangement tableaux of shape equal to the given partit....
St000509 Integer partitions ⟶ ℤ
The diagonal index (content) of a partition.
St000510 Integer partitions ⟶ ℤ
The number of invariant oriented cycles when acting with a permutation of given c....
St000511 Integer partitions ⟶ ℤ
The number of invariant subsets when acting with a permutation of given cycle typ....
St000512 Integer partitions ⟶ ℤ
The number of invariant subsets of size 3 when acting with a permutation of given....
St000513 Integer partitions ⟶ ℤ
The number of invariant subsets of size 2 when acting with a permutation of given....
St000514 Integer partitions ⟶ ℤ
The number of invariant simple graphs when acting with a permutation of given cyc....
St000515 Integer partitions ⟶ ℤ
The number of invariant set partitions when acting with a permutation of given cy....
St000517 Integer partitions ⟶ ℤ
The Kreweras number of an integer partition.
St000531 Integer partitions ⟶ ℤ
The number of ways to place as many non-attacking rooks as possible on a Ferrers ....
St000532 Integer partitions ⟶ ℤ
The total number of rook placements on a Ferrers board.
St000533 Integer partitions ⟶ ℤ
The maximal number of non-attacking rooks on a Ferrers shape.
St000547 Integer partitions ⟶ ℤ
The number of even non-empty partial sums of an integer partition.
St000548 Integer partitions ⟶ ℤ
The number of different non-empty partial sums of an integer partition.
St000549 Integer partitions ⟶ ℤ
The number of odd partial sums of an integer partition.
St000566 Integer partitions ⟶ ℤ
The number of ways to select a row of a Ferrers shape and two cells in this row.
St000567 Integer partitions ⟶ ℤ
The sum of the products of all pairs of parts.
St000618 Integer partitions ⟶ ℤ
The number of self-evacuating tableaux of given shape.
St000620 Integer partitions ⟶ ℤ
The number of standard tableaux of shape equal to the given partition such that t....
St000621 Integer partitions ⟶ ℤ
The number of standard tableaux of shape equal to the given partition such that t....
St000644 Integer partitions ⟶ ℤ
The number of graphs with given frequency partition.
St000667 Integer partitions ⟶ ℤ
The greatest common divisor of the parts of the partition.
St000668 Integer partitions ⟶ ℤ
The least common multiple of the parts of the partition.
St000681 Integer partitions ⟶ ℤ
The Grundy value of Chomp on Ferrers diagrams.
St000697 Integer partitions ⟶ ℤ
The number of 3-rim hooks removed from an integer partition to obtain its associa....
St000698 Integer partitions ⟶ ℤ
The number of 2-rim hooks removed from an integer partition to obtain its associa....
St000704 Integer partitions ⟶ ℤ
The number of semistandard tableaux on a given integer partition with minimal max....
St000705 Integer partitions ⟶ ℤ
The number of semistandard tableaux on a given integer partition of n with maxima....
St000706 Integer partitions ⟶ ℤ
The product of the factorials of the multiplicities of an integer partition.
St000707 Integer partitions ⟶ ℤ
The product of the factorials of the parts.
St000708 Integer partitions ⟶ ℤ
The product of the parts of an integer partition.
St000712 Integer partitions ⟶ ℤ
The number of semistandard Young tableau of given shape, with entries at most 4.
St000713 Integer partitions ⟶ ℤ
The dimension of the irreducible representation of Sp(4) labelled by an integer p....
St000714 Integer partitions ⟶ ℤ
The number of semistandard Young tableau of given shape, with entries at most 2.
St000715 Integer partitions ⟶ ℤ
The number of semistandard Young tableaux of given shape and entries at most 3.
St000716 Integer partitions ⟶ ℤ
The dimension of the irreducible representation of Sp(6) labelled by an integer p....
St000749 Integer partitions ⟶ ℤ
The smallest integer d such that the restriction of the representation correspond....
St000752 Integer partitions ⟶ ℤ
The Grundy value for the game 'Couples are forever' on an integer partition.
St000755 Integer partitions ⟶ ℤ
The number of real roots of the characteristic polynomial of a linear recurrence ....
St000759 Integer partitions ⟶ ℤ
The smallest missing part in an integer partition.
St000770 Integer partitions ⟶ ℤ
The major index of an integer partition when read from bottom to top.
St000781 Integer partitions ⟶ ℤ
The number of proper colouring schemes of a Ferrers diagram.
St000783 Integer partitions ⟶ ℤ
The maximal number of occurrences of a colour in a proper colouring of a Ferrers ....
St000784 Integer partitions ⟶ ℤ
The maximum of the length and the largest part of the integer partition.
St000810 Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of ....
St000811 Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of ....
St000812 Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of ....
St000813 Integer partitions ⟶ ℤ
The number of zero-one matrices with weakly decreasing column sums and row sums g....
St000814 Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of ....
St000815 Integer partitions ⟶ ℤ
The number of semistandard Young tableaux of partition weight of given shape.
St000835 Integer partitions ⟶ ℤ
The minimal difference in size when partitioning the integer partition into two s....
St000867 Integer partitions ⟶ ℤ
The sum of the hook lengths in the first column of an integer partition.
St000869 Integer partitions ⟶ ℤ
The sum of the hook lengths of an integer partition.
St000870 Integer partitions ⟶ ℤ
The product of the hook lengths of the diagonal cells in an integer partition.
St000897 Integer partitions ⟶ ℤ
The number of different multiplicities of parts of an integer partition.
St000901 Integer partitions ⟶ ℤ
The cube of the number of standard Young tableaux with shape given by the partiti....
St000913 Integer partitions ⟶ ℤ
The number of ways to refine the partition into singletons.
St000927 Integer partitions ⟶ ℤ
The alternating sum of the coefficients of the character polynomial of an integer....
St000928 Integer partitions ⟶ ℤ
The sum of the coefficients of the character polynomial of an integer partition.
St000929 Integer partitions ⟶ ℤ
The constant term of the character polynomial of an integer partition.
St000933 Integer partitions ⟶ ℤ
The number of multipartitions of sizes given by an integer partition.
St000934 Integer partitions ⟶ ℤ
The 2-degree of an integer partition.
St000935 Integer partitions ⟶ ℤ
The number of ordered refinements of an integer partition.
St000936 Integer partitions ⟶ ℤ
The number of even values of the symmetric group character corresponding to the p....
St000937 Integer partitions ⟶ ℤ
The number of positive values of the symmetric group character corresponding to t....
St000938 Integer partitions ⟶ ℤ
The number of zeros of the symmetric group character corresponding to the partiti....
St000939 Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is p....
St000940 Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is z....
St000941 Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is e....
St000944 Integer partitions ⟶ ℤ
The 3-degree of an integer partition.
St000992 Integer partitions ⟶ ℤ
The alternating sum of the parts of an integer partition.
St000993 Integer partitions ⟶ ℤ
The multiplicity of the largest part of an integer partition.
St000995 Integer partitions ⟶ ℤ
The largest even part of an integer partition.
St000997 Integer partitions ⟶ ℤ
The even-odd crank of an integer partition.
St001055 Integer partitions ⟶ ℤ
The Grundy value for the game of removing cells of a row in an integer partition.....
St001091 Integer partitions ⟶ ℤ
The number of parts in an integer partition whose next smaller part has the same ....
St001092 Integer partitions ⟶ ℤ
The number of distinct even parts of a partition.
St001097 Integer partitions ⟶ ℤ
The coefficient of the monomial symmetric function indexed by the partition in th....
St001098 Integer partitions ⟶ ℤ
The coefficient times the product of the factorials of the parts of the monomial ....
St001099 Integer partitions ⟶ ℤ
The coefficient times the product of the factorials of the parts of the monomial ....
St001100 Integer partitions ⟶ ℤ
The coefficient times the product of the factorials of the parts of the monomial ....
St001101 Integer partitions ⟶ ℤ
The coefficient times the product of the factorials of the parts of the monomial ....
St001103 Integer partitions ⟶ ℤ
The number of words with multiplicities of the letters given by the partition, av....
St001121 Integer partitions ⟶ ℤ
The multiplicity of the irreducible representation indexed by the partition in th....
St001122 Integer partitions ⟶ ℤ
The multiplicity of the sign representation in the Kronecker square corresponding....
St001123 Integer partitions ⟶ ℤ
The multiplicity of the dual of the standard representation in the Kronecker squa....
St001124 Integer partitions ⟶ ℤ
The multiplicity of the standard representation in the Kronecker square correspon....
St001127 Integer partitions ⟶ ℤ
The sum of the squares of the parts of a partition.
St001128 Integer partitions ⟶ ℤ
The exponens consonantiae of a partition.
St001129 Integer partitions ⟶ ℤ
The product of the squares of the parts of a partition.
St001175 Integer partitions ⟶ ℤ
The size of a partition minus the hook length of the base cell.
St001176 Integer partitions ⟶ ℤ
The size of a partition minus its first part.
St001177 Integer partitions ⟶ ℤ
Twice the mean value of the major index among all standard Young tableaux of a pa....
St001178 Integer partitions ⟶ ℤ
Twelve times the variance of the major index among all standard Young tableaux of....
   Ordered trees (23 statistics) # tree like structures # Catalan objects # graph like objects
St000084 Ordered trees ⟶ ℤ
The number of subtrees.
St000085 Ordered trees ⟶ ℤ
The number of linear extensions of the tree.
St000094 Ordered trees ⟶ ℤ
The depth of an ordered tree.
St000166 Ordered trees ⟶ ℤ
The depth minus 1 of an ordered tree.
St000167 Ordered trees ⟶ ℤ
The number of leaves of an ordered tree.
St000168 Ordered trees ⟶ ℤ
The number of internal nodes of an ordered tree.
St000328 Ordered trees ⟶ ℤ
The maximum number of child nodes in a tree.
St000397 Ordered trees ⟶ ℤ
The Strahler number of a rooted tree.
St000400 Ordered trees ⟶ ℤ
The path length of an ordered tree.
St000410 Ordered trees ⟶ ℤ
The tree factorial of an ordered tree.
St000413 Ordered trees ⟶ ℤ
The number of ordered trees with the same underlying unordered tree.
St000415 Ordered trees ⟶ ℤ
The size of the automorphism group of the rooted tree underlying the ordered tree....
St000416 Ordered trees ⟶ ℤ
The number of inequivalent increasing trees of an ordered tree.
St000417 Ordered trees ⟶ ℤ
The size of the automorphism group of the ordered tree.
St000521 Ordered trees ⟶ ℤ
The number of distinct subtrees of an ordered tree.
St000522 Ordered trees ⟶ ℤ
The number of 1-protected nodes of a rooted tree.
St000523 Ordered trees ⟶ ℤ
The number of 2-protected nodes of a rooted tree.
St000679 Ordered trees ⟶ ℤ
The pruning number of an ordered tree.
St000700 Ordered trees ⟶ ℤ
The protection number of an ordered tree.
St000973 Ordered trees ⟶ ℤ
The length of the boundary of an ordered tree.
St000974 Ordered trees ⟶ ℤ
The length of the trunk of an ordered tree.
St000975 Ordered trees ⟶ ℤ
The length of the boundary minus the length of the trunk of an ordered tree.
St001058 Ordered trees ⟶ ℤ
The breadth of the ordered tree.
   Parking functions (10 statistics) # word like objects # path like objects
St000135 Parking functions ⟶ ℤ
The number of lucky cars of the parking function.
St000136 Parking functions ⟶ ℤ
The dinv of a parking function.
St000165 Parking functions ⟶ ℤ
Sum of the entries.
St000188 Parking functions ⟶ ℤ
The area of the Dyck path corresponding to a parking function.
St000194 Parking functions ⟶ ℤ
The number of primary dinversion pairs of a labelled dyck path corresponding to a....
St000195 Parking functions ⟶ ℤ
The number of secondary dinversion pairs of the dyck path corresponding to a park....
St000540 Parking functions ⟶ ℤ
The sum of the entries of a parking function minus its length.
St000942 Parking functions ⟶ ℤ
The number of critical left to right maxima of the parking functions.
St000943 Parking functions ⟶ ℤ
The number of spots the most unlucky car had to go further in a parking function.....
St001209 Parking functions ⟶ ℤ
The pmaj statistic of a parking function.
   Perfect matchings (38 statistics) # graph like objects
St000041 Perfect matchings ⟶ ℤ
The number of nestings of a perfect matching.
St000042 Perfect matchings ⟶ ℤ
The number of crossings of a perfect matching.
St000043 Perfect matchings ⟶ ℤ
The number of crossings plus two-nestings of a perfect matching.
St000044 Perfect matchings ⟶ ℤ
The number of vertices of the unicellular map given by a perfect matching.
St000164 Perfect matchings ⟶ ℤ
The number of short pairs.
St000719 Perfect matchings ⟶ ℤ
The number of alignments in a perfect matching.
St000720 Perfect matchings ⟶ ℤ
The size of the largest partition in the oscillating tableau corresponding to the....
St000721 Perfect matchings ⟶ ℤ
The sum of the partition sizes in the oscillating tableau corresponding to a perf....
St000746 Perfect matchings ⟶ ℤ
The number of pairs with odd minimum in a perfect matching.
St000754 Perfect matchings ⟶ ℤ
The Grundy value for the game of removing nestings in a perfect matching.
St000780 Perfect matchings ⟶ ℤ
The size of the orbit under rotation of a perfect matching.
St000782 Perfect matchings ⟶ ℤ
The indicator function of whether a given perfect matching is an L & P matching.
St000787 Perfect matchings ⟶ ℤ
The number of flips required to make a perfect matching noncrossing.
St000788 Perfect matchings ⟶ ℤ
The number of nesting-similar perfect matchings of a perfect matching.
St000789 Perfect matchings ⟶ ℤ
The number of crossing-similar perfect matchings of a perfect matching.
St000819 Perfect matchings ⟶ ℤ
The propagating number of a perfect matching.
St000838 Perfect matchings ⟶ ℤ
The number of terminal right-hand endpoints when the vertices are written in orde....
St000840 Perfect matchings ⟶ ℤ
The number of closers smaller than the largest opener in a perfect matching.
St000841 Perfect matchings ⟶ ℤ
The largest opener of a perfect matching.
St000843 Perfect matchings ⟶ ℤ
The decomposition number of a perfect matching.
St000924 Perfect matchings ⟶ ℤ
The number of topologically connected components of a perfect matching.
St000945 Perfect matchings ⟶ ℤ
The number of matchings in the dihedral orbit of a perfect matching.
St001040 Perfect matchings ⟶ ℤ
The depth of the decreasing labelled binary unordered tree associated with the pe....
St001041 Perfect matchings ⟶ ℤ
The depth of the label 1 in the decreasing labelled binary unordered tree associa....
St001042 Perfect matchings ⟶ ℤ
The size of the automorphism group of the leaf labelled binary unordered tree ass....
St001043 Perfect matchings ⟶ ℤ
The depth of the leaf closest to the root in the binary unordered tree associated....
St001044 Perfect matchings ⟶ ℤ
The number of pairs whose larger element is at most one more than half the size o....
St001045 Perfect matchings ⟶ ℤ
The number of leaves in the subtree not containing one in the decreasing labelled....
St001046 Perfect matchings ⟶ ℤ
The maximal number of arcs nesting a given arc of a perfect matching.
St001047 Perfect matchings ⟶ ℤ
The maximal number of arcs crossing a given arc of a perfect matching.
St001048 Perfect matchings ⟶ ℤ
The number of leaves in the subtree containing 1 in the decreasing labelled binar....
St001049 Perfect matchings ⟶ ℤ
The smallest label in the subtree not containing 1 in the decreasing labelled bin....
St001131 Perfect matchings ⟶ ℤ
The number of trivial trees on the path to label one in the decreasing labelled b....
St001132 Perfect matchings ⟶ ℤ
The number of leaves in the subtree whose sister has label 1 in the decreasing la....
St001133 Perfect matchings ⟶ ℤ
The smallest label in the subtree rooted at the sister of 1 in the decreasing lab....
St001134 Perfect matchings ⟶ ℤ
The largest label in the subtree rooted at the sister of 1 in the leaf labelled b....
St001136 Perfect matchings ⟶ ℤ
The largest label with larger sister in the leaf labelled binary unordered tree a....
St001152 Perfect matchings ⟶ ℤ
The number of pairs with even minimum in a perfect matching.
   Permutations (295 statistics) # word like objects
St000001 Permutations ⟶ ℤ
The number of ways to write a permutation as a minimal length product of simple t....
St000002 Permutations ⟶ ℤ
The number of occurrences of the pattern 123 in a permutation.
St000004 Permutations ⟶ ℤ
The major index of a permutation.
St000007 Permutations ⟶ ℤ
The number of saliances of the permutation.
St000018 Permutations ⟶ ℤ
The number of inversions of a permutation.
St000019 Permutations ⟶ ℤ
The cardinality of the complement of the connectivity set.
St000020 Permutations ⟶ ℤ
The rank of the permutation.
St000021 Permutations ⟶ ℤ
The number of descents of a permutation.
St000022 Permutations ⟶ ℤ
The number of fixed points of a permutation.
St000023 Permutations ⟶ ℤ
The number of inner peaks of a permutation.
St000028 Permutations ⟶ ℤ
The number of stack-sorts needed to sort a permutation.
St000029 Permutations ⟶ ℤ
The depth of a permutation.
St000030 Permutations ⟶ ℤ
The sum of the descent differences of a permutations.
St000031 Permutations ⟶ ℤ
The number of cycles in the cycle decomposition of a permutation.
St000033 Permutations ⟶ ℤ
The number of permutations greater than or equal to the given permutation in (str....
St000034 Permutations ⟶ ℤ
The maximum defect over any reduced expression for a permutation and any subexpre....
St000035 Permutations ⟶ ℤ
The number of left outer peaks of a permutation.
St000036 Permutations ⟶ ℤ
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by th....
St000037 Permutations ⟶ ℤ
The sign of a permutation.
St000039 Permutations ⟶ ℤ
The number of crossings of a permutation.
St000040 Permutations ⟶ ℤ
The number of regions of inversion arrangement of a permutation.
St000054 Permutations ⟶ ℤ
The first entry of the permutation.
St000055 Permutations ⟶ ℤ
The inversion sum of a permutation.
St000056 Permutations ⟶ ℤ
The decomposition (or block) number of a permutation.
St000058 Permutations ⟶ ℤ
The order of a permutation.
St000060 Permutations ⟶ ℤ
The greater neighbor of the maximum.
St000062 Permutations ⟶ ℤ
The length of the longest increasing subsequence of the permutation.
St000064 Permutations ⟶ ℤ
The number of one-box pattern of a permutation.
St000078 Permutations ⟶ ℤ
The number of alternating sign matrices whose left key is the permutation.
St000092 Permutations ⟶ ℤ
The number of outer peaks of a permutation.
St000099 Permutations ⟶ ℤ
The number of valleys of a permutation, including the boundary.
St000109 Permutations ⟶ ℤ
The number of elements less than or equal to the given element in Bruhat order.
St000110 Permutations ⟶ ℤ
The number of permutations less than or equal to given permutation in left weak o....
St000111 Permutations ⟶ ℤ
The sum of the descent tops (or Genocchi descents) of a permutation.
St000119 Permutations ⟶ ℤ
The number of occurrences of the pattern 321 in a permutation.
St000123 Permutations ⟶ ℤ
The difference in Coxeter length of a permutation and its image under the Simion-....
St000124 Permutations ⟶ ℤ
The cardinality of the preimage of the Simion-Schmidt map.
St000133 Permutations ⟶ ℤ
The "bounce" of a permutation.
St000141 Permutations ⟶ ℤ
The maximum drop size of a permutation.
St000153 Permutations ⟶ ℤ
The number of adjacent cycles of a permutation.
St000154 Permutations ⟶ ℤ
The sum of the descent bottoms of a permutation.
St000155 Permutations ⟶ ℤ
The number of exceedances (also excedences) of a permutation.
St000156 Permutations ⟶ ℤ
The Denert index of a permutation.
St000162 Permutations ⟶ ℤ
The number of nontrivial cycles of a permutation $\pi$ in its cycle decomposition.
St000209 Permutations ⟶ ℤ
Maximum difference of elements in cycles.
St000210 Permutations ⟶ ℤ
Minimum over maximum difference of elements in cycles.
St000213 Permutations ⟶ ℤ
The number of weak exceedances (also weak excedences) of a permutation.
St000214 Permutations ⟶ ℤ
The number of adjacencies of a permutation.
St000215 Permutations ⟶ ℤ
The number of adjacencies of a permutation, zero appended.
St000216 Permutations ⟶ ℤ
The absolute length of a permutation.
St000217 Permutations ⟶ ℤ
The number of occurrences of the pattern 312 in a permutation.
St000218 Permutations ⟶ ℤ
The number of occurrences of the pattern 213 in a permutation.
St000219 Permutations ⟶ ℤ
The number of occurrences of the pattern 231 in a permutation.
St000220 Permutations ⟶ ℤ
The number of occurrences of the pattern 132 in a permutation.
St000221 Permutations ⟶ ℤ
The number of strong fixed points of a permutation.
St000222 Permutations ⟶ ℤ
The number of alignments in the permutation.
St000223 Permutations ⟶ ℤ
The number of nestings in the permutation.
St000224 Permutations ⟶ ℤ
The sorting index of a permutation.
St000226 Permutations ⟶ ℤ
The convexity of a permutation.
St000234 Permutations ⟶ ℤ
The number of global ascents of a permutation.
St000235 Permutations ⟶ ℤ
The number of indices $i$ such that $\pi_i \neq i+1$ considered cyclically.
St000236 Permutations ⟶ ℤ
The number of indices $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
St000237 Permutations ⟶ ℤ
The number of small exceedances.
St000238 Permutations ⟶ ℤ
The number of indices $i$ such that $\pi_i \notin \{i,i+1\}$.
St000239 Permutations ⟶ ℤ
The number of indices $i$ such that $\pi_i \in \{i,i+1\}$.
St000240 Permutations ⟶ ℤ
The number of indices $i$ for which $\pi_i \neq i+1$.
St000241 Permutations ⟶ ℤ
The number of indices $i$ such that $\pi_i = i+1$ considered cyclically.
St000242 Permutations ⟶ ℤ
The number of indices $i$ such that $\pi_i \notin \{ i,i+1 \}$ considered cyclically.....
St000243 Permutations ⟶ ℤ
The number of cyclic valleys and cyclic peaks of a permutation.
St000245 Permutations ⟶ ℤ
The number of ascents of a permutation.
St000246 Permutations ⟶ ℤ
The number of non-inversions of a permutation.
St000255 Permutations ⟶ ℤ
The number of reduced Kogan faces with the permutation as type.
St000279 Permutations ⟶ ℤ
The size of the preimage of the map 'cycle-as-one-line notation' from Permutation....
St000280 Permutations ⟶ ℤ
The size of the preimage of the map 'to labelling permutation' from Parking funct....
St000304 Permutations ⟶ ℤ
The load of a permutation.
St000305 Permutations ⟶ ℤ
The inverse major index of a permutation.
St000308 Permutations ⟶ ℤ
The height of the tree associated to a permutation.
St000314 Permutations ⟶ ℤ
The number of left-to-right-maxima of a permutation.
St000316 Permutations ⟶ ℤ
The number of non-left-to-right-maxima of a permutation.
St000317 Permutations ⟶ ℤ
The cycle descent number of a permutation.
St000324 Permutations ⟶ ℤ
The shape of the tree associated to a permutation.
St000325 Permutations ⟶ ℤ
The width of the tree associated to a permutation.
St000333 Permutations ⟶ ℤ
The dez statistic, the number of descents of a permutation after replacing fixed ....
St000334 Permutations ⟶ ℤ
The maz index, the major index of a permutation after replacing fixed points by z....
St000337 Permutations ⟶ ℤ
The lec statistic, the sum of the inversion numbers of the hook factors of a perm....
St000338 Permutations ⟶ ℤ
The number of pixed points of a permutation.
St000339 Permutations ⟶ ℤ
The maf index of a permutation.
St000341 Permutations ⟶ ℤ
The non-inversion sum of a permutation.
St000342 Permutations ⟶ ℤ
The cosine of a permutation.
St000352 Permutations ⟶ ℤ
The Elizalde-Pak rank of a permutation.
St000353 Permutations ⟶ ℤ
The number of inner valleys of a permutation.
St000354 Permutations ⟶ ℤ
The number of recoils of a permutation.
St000355 Permutations ⟶ ℤ
The number of occurrences of the pattern 21-3.
St000356 Permutations ⟶ ℤ
The number of occurrences of the pattern 13-2.
St000357 Permutations ⟶ ℤ
The number of occurrences of the pattern 12-3.
St000358 Permutations ⟶ ℤ
The number of occurrences of the pattern 31-2.
St000359 Permutations ⟶ ℤ
The number of occurrences of the pattern 23-1.
St000360 Permutations ⟶ ℤ
The number of occurrences of the pattern 32-1.
St000365 Permutations ⟶ ℤ
The number of double ascents of a permutation.
St000366 Permutations ⟶ ℤ
The number of double descents of a permutation.
St000367 Permutations ⟶ ℤ
The number of simsun double descents of a permutation.
St000371 Permutations ⟶ ℤ
The number of mid points of decreasing subsequences of length 3 in a permutation.....
St000372 Permutations ⟶ ℤ
The number of mid points of increasing subsequences of length 3 in a permutation.....
St000373 Permutations ⟶ ℤ
The number of weak exceedences of a permutation that are also mid-points of a dec....
St000374 Permutations ⟶ ℤ
The number of exclusive right-to-left minima of a permutation.
St000375 Permutations ⟶ ℤ
The number of non weak exceedences of a permutation that are mid-points of a decr....
St000401 Permutations ⟶ ℤ
The size of the symmetry class of a permutation.
St000402 Permutations ⟶ ℤ
Half the size of the symmetry class of a permutation.
St000404 Permutations ⟶ ℤ
The number of occurrences of the pattern 3241 or of the pattern 4231 in a permuta....
St000405 Permutations ⟶ ℤ
The number of occurrences of the pattern 1324 in a permutation.
St000406 Permutations ⟶ ℤ
The number of occurrences of the pattern 3241 in a permutation.
St000407 Permutations ⟶ ℤ
The number of occurrences of the pattern 2143 in a permutation.
St000408 Permutations ⟶ ℤ
The number of occurrences of the pattern 4231 in a permutation.
St000423 Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 132 in a permutati....
St000424 Permutations ⟶ ℤ
The number of occurrences of the pattern 132 or of the pattern 231 in a permutati....
St000425 Permutations ⟶ ℤ
The number of occurrences of the pattern 132 or of the pattern 213 in a permutati....
St000426 Permutations ⟶ ℤ
The number of occurrences of the pattern 132 or of the pattern 312 in a permutati....
St000427 Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 231 in a permutati....
St000428 Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 213 in a permutati....
St000429 Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 321 in a permutati....
St000430 Permutations ⟶ ℤ
The number of occurrences of the pattern 123 or of the pattern 312 in a permutati....
St000431 Permutations ⟶ ℤ
The number of occurrences of the pattern 213 or of the pattern 321 in a permutati....
St000432 Permutations ⟶ ℤ
The number of occurrences of the pattern 231 or of the pattern 312 in a permutati....
St000433 Permutations ⟶ ℤ
The number of occurrences of the pattern 132 or of the pattern 321 in a permutati....
St000434 Permutations ⟶ ℤ
The number of occurrences of the pattern 213 or of the pattern 312 in a permutati....
St000435 Permutations ⟶ ℤ
The number of occurrences of the pattern 213 or of the pattern 231 in a permutati....
St000436 Permutations ⟶ ℤ
The number of occurrences of the pattern 231 or of the pattern 321 in a permutati....
St000437 Permutations ⟶ ℤ
The number of occurrences of the pattern 312 or of the pattern 321 in a permutati....
St000440 Permutations ⟶ ℤ
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permuta....
St000441 Permutations ⟶ ℤ
The number of successions of a permutation.
St000446 Permutations ⟶ ℤ
The disorder of a permutation.
St000451 Permutations ⟶ ℤ
The length of the longest pattern of the form k 1 2.
St000457 Permutations ⟶ ℤ
The number of occurrences of one of the patterns 132, 213 or 321 in a permutation....
St000458 Permutations ⟶ ℤ
The number of permutations obtained by switching adjacencies or successions.
St000461 Permutations ⟶ ℤ
The rix statistic of a permutation.
St000462 Permutations ⟶ ℤ
The major index minus the number of excedences of a permutation.
St000463 Permutations ⟶ ℤ
The number of admissible inversions of a permutation.
St000470 Permutations ⟶ ℤ
The number of runs in a permutation.
St000471 Permutations ⟶ ℤ
The sum of the ascent tops of a permutation.
St000472 Permutations ⟶ ℤ
The sum of the ascent bottoms of a permutation.
St000483 Permutations ⟶ ℤ
The number of times a permutation switches from increasing to decreasing or decre....
St000484 Permutations ⟶ ℤ
The sum of St000483 over all subsequences of length at least three.
St000485 Permutations ⟶ ℤ
The length of the longest cycle of a permutation.
St000486 Permutations ⟶ ℤ
The number of cycles of length at least 3 of a permutation.
St000487 Permutations ⟶ ℤ
The length of the shortest cycle of a permutation.
St000488 Permutations ⟶ ℤ
The number of cycles of a permutation of length at most 2.
St000489 Permutations ⟶ ℤ
The number of cycles of a permutation of length at most 3.
St000494 Permutations ⟶ ℤ
The number of inversions of distance at most 3 of a permutation.
St000495 Permutations ⟶ ℤ
The number of inversions of distance at most 2 of a permutation.
St000500 Permutations ⟶ ℤ
Eigenvalues of the random-to-random operator acting on the regular representation....
St000501 Permutations ⟶ ℤ
The size of the first part in the decomposition of a permutation.
St000516 Permutations ⟶ ℤ
The number of stretching pairs of a permutation.
St000520 Permutations ⟶ ℤ
The number of patterns in a permutation.
St000530 Permutations ⟶ ℤ
The number of permutations with the same descent word as the given permutation.
St000534 Permutations ⟶ ℤ
The number of 2-rises of a permutation.
St000538 Permutations ⟶ ℤ
The number of even inversions of a permutation.
St000539 Permutations ⟶ ℤ
The number of odd inversions of a permutation.
St000541 Permutations ⟶ ℤ
The number of indices greater than or equal to 2 of a permutation such that all s....
St000542 Permutations ⟶ ℤ
The number of left-to-right-minima of a permutation.
St000545 Permutations ⟶ ℤ
The number of parabolic double cosets with minimal element being the given permut....
St000546 Permutations ⟶ ℤ
The number of global descents of a permutation.
St000570 Permutations ⟶ ℤ
The Edelman-Greene number of a permutation.
St000616 Permutations ⟶ ℤ
The inversion index of a permutation.
St000619 Permutations ⟶ ℤ
The number of cyclic descents of a permutation.
St000622 Permutations ⟶ ℤ
The number of occurrences of the patterns 2143 or 4231 in a permutation.
St000623 Permutations ⟶ ℤ
The number of occurrences of the pattern 52341 in a permutation.
St000624 Permutations ⟶ ℤ
The normalized sum of the minimal distances to a greater element.
St000625 Permutations ⟶ ℤ
The sum of the minimal distances to a greater element.
St000638 Permutations ⟶ ℤ
The number of up-down runs of a permutation.
St000646 Permutations ⟶ ℤ
The number of big ascents of a permutation.
St000647 Permutations ⟶ ℤ
The number of big descents of a permutation.
St000648 Permutations ⟶ ℤ
The number of 2-excedences of a permutation.
St000649 Permutations ⟶ ℤ
The number of 3-excedences of a permutation.
St000650 Permutations ⟶ ℤ
The number of 3-rises of a permutation.
St000651 Permutations ⟶ ℤ
The maximal size of a rise in a permutation.
St000652 Permutations ⟶ ℤ
The maximal difference between successive positions of a permutation.
St000653 Permutations ⟶ ℤ
The last descent of a permutation.
St000654 Permutations ⟶ ℤ
The first descent of a permutation.
St000662 Permutations ⟶ ℤ
The staircase size of the code of a permutation.
St000663 Permutations ⟶ ℤ
The number of right floats of a permutation.
St000664 Permutations ⟶ ℤ
The number of right ropes of a permutation.
St000665 Permutations ⟶ ℤ
The number of rafts of a permutation.
St000666 Permutations ⟶ ℤ
The number of right tethers of a permutation.
St000669 Permutations ⟶ ℤ
The number of permutations obtained by switching ascents or descents of size 2.
St000670 Permutations ⟶ ℤ
The reversal length of a permutation.
St000672 Permutations ⟶ ℤ
The number of minimal elements in Bruhat order not less than the permutation.
St000673 Permutations ⟶ ℤ
The size of the support of a permutation.
St000677 Permutations ⟶ ℤ
The standardized bi-alternating inversion number of a permutation.
St000690 Permutations ⟶ ℤ
The size of the conjugacy class of a permutation.
St000692 Permutations ⟶ ℤ
Babson and Steingrímsson's statistic stat of a permutation.
St000694 Permutations ⟶ ℤ
The number of affine bounded permutations that project to a given permutation.
St000696 Permutations ⟶ ℤ
The number of cycles in the breakpoint graph of a permutation.
St000702 Permutations ⟶ ℤ
The number of weak deficiencies of a permutation.
St000703 Permutations ⟶ ℤ
The number of deficiencies of a permutation.
St000709 Permutations ⟶ ℤ
The number of occurrences of 14-2-3 or 14-3-2.
St000710 Permutations ⟶ ℤ
The number of big deficiencies of a permutation.
St000711 Permutations ⟶ ℤ
The number of big exceedences of a permutation.
St000724 Permutations ⟶ ℤ
The label of the leaf of the path following the smaller label in the increasing b....
St000725 Permutations ⟶ ℤ
The smallest label of a leaf of the increasing binary tree associated to a permut....
St000726 Permutations ⟶ ℤ
The normalized sum of the leaf labels of the increasing binary tree associated to....
St000727 Permutations ⟶ ℤ
The largest label of a leaf in the binary search tree associated with the permuta....
St000731 Permutations ⟶ ℤ
The number of double exceedences of a permutation.
St000732 Permutations ⟶ ℤ
The number of double deficiencies of a permutation.
St000740 Permutations ⟶ ℤ
The last entry of a permutation.
St000742 Permutations ⟶ ℤ
The number of big ascents of a permutation after adding the value $\pi(0) = 0$.
St000750 Permutations ⟶ ℤ
The number of occurrences of the pattern 4213 in a permutation.
St000751 Permutations ⟶ ℤ
The number of occurrences of either of the pattern 2143 or 2143 in a permutation.....
St000756 Permutations ⟶ ℤ
The sum of the positions of the left to right maxima of a permutation.
St000779 Permutations ⟶ ℤ
The tier of a permutation.
St000794 Permutations ⟶ ℤ
The mak of a permutation.
St000795 Permutations ⟶ ℤ
The mad of a permutation.
St000796 Permutations ⟶ ℤ
The stat' of a permutation.
St000797 Permutations ⟶ ℤ
The stat`` of a permutation.
St000798 Permutations ⟶ ℤ
The makl of a permutation.
St000799 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |213 in a permutation.
St000800 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |231 in a permutation.
St000801 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |312 in a permutation.
St000802 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |321 in a permutation.
St000803 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |132 in a permutation.
St000804 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |123 in a permutation.
St000809 Permutations ⟶ ℤ
The reduced reflection length of the permutation.
St000824 Permutations ⟶ ℤ
The sum of the number of descents and the number of recoils of a permutation.
St000825 Permutations ⟶ ℤ
The sum of the major and the inverse major index of a permutation.
St000828 Permutations ⟶ ℤ
The spearman's rho of a permutation and the identity permutation.
St000829 Permutations ⟶ ℤ
The Ulam distance of a permutation to the identity permutation.
St000830 Permutations ⟶ ℤ
The total displacement of a permutation.
St000831 Permutations ⟶ ℤ
The number of indices that are either descents or recoils.
St000832 Permutations ⟶ ℤ
The number of permutations obtained by reversing blocks of three consecutive numb....
St000833 Permutations ⟶ ℤ
The comajor index of a permutation.
St000834 Permutations ⟶ ℤ
The number of right outer peaks of a permutation.
St000836 Permutations ⟶ ℤ
The number of descents of distance 2 of a permutation.
St000837 Permutations ⟶ ℤ
The number of ascents of distance 2 of a permutation.
St000842 Permutations ⟶ ℤ
The breadth of a permutation.
St000844 Permutations ⟶ ℤ
The size of the largest block in the direct sum decomposition of a permutation.
St000862 Permutations ⟶ ℤ
The number of parts of the shifted shape of a permutation.
St000863 Permutations ⟶ ℤ
The length of the first row of the shifted shape of a permutation.
St000864 Permutations ⟶ ℤ
The number of circled entries of the shifted recording tableau of a permutation.
St000866 Permutations ⟶ ℤ
The number of admissible inversions of a permutation in the sense of Shareshian-W....
St000868 Permutations ⟶ ℤ
The aid statistic in the sense of Shareshian-Wachs.
St000871 Permutations ⟶ ℤ
The number of very big ascents of a permutation.
St000872 Permutations ⟶ ℤ
The number of very big descents of a permutation.
St000873 Permutations ⟶ ℤ
The aix statistic of a permutation.
St000879 Permutations ⟶ ℤ
The number of long braid edges in the graph of braid moves of a permutation.
St000880 Permutations ⟶ ℤ
The number of connected components of long braid edges in the graph of braid move....
St000881 Permutations ⟶ ℤ
The number of short braid edges in the graph of braid moves of a permutation.
St000882 Permutations ⟶ ℤ
The number of connected components of short braid edges in the graph of braid mov....
St000883 Permutations ⟶ ℤ
The number of longest increasing subsequences of a permutation.
St000884 Permutations ⟶ ℤ
The number of isolated descents of a permutation.
St000886 Permutations ⟶ ℤ
The number of permutations with the same antidiagonal sums.
St000887 Permutations ⟶ ℤ
The maximal number of nonzero entries on a diagonal of a permutation matrix.
St000891 Permutations ⟶ ℤ
The number of distinct diagonal sums of a permutation matrix.
St000923 Permutations ⟶ ℤ
The minimal number with no two order isomorphic substrings of this length in a pe....
St000956 Permutations ⟶ ℤ
The maximal displacement of a permutation.
St000957 Permutations ⟶ ℤ
The number of Bruhat lower covers of a permutation.
St000958 Permutations ⟶ ℤ
The number of Bruhat factorizations of a permutation.
St000959 Permutations ⟶ ℤ
The number of strong Bruhat factorizations of a permutation.
St000961 Permutations ⟶ ℤ
The shifted major index of a permutation.
St000962 Permutations ⟶ ℤ
The 3-shifted major index of a permutation.
St000963 Permutations ⟶ ℤ
The 2-shifted major index of a permutation.
St000988 Permutations ⟶ ℤ
The orbit size of a permutation under Foata's bijection.
St000989 Permutations ⟶ ℤ
The number of final rises of a permutation.
St000990 Permutations ⟶ ℤ
The first ascent of a permutation.
St000991 Permutations ⟶ ℤ
The number of right-to-left minima of a permutation.
St000994 Permutations ⟶ ℤ
The number of cycle peaks and the number of cycle valleys of a permutation.
St000996 Permutations ⟶ ℤ
The number of exclusive left-to-right maxima of a permutation.
St001004 Permutations ⟶ ℤ
The number of indices that are either left-to-right maxima or right-to-left minim....
St001005 Permutations ⟶ ℤ
The number of indices for a permutation that are either left-to-right maxima or r....
St001052 Permutations ⟶ ℤ
The length of the exterior of a permutation.
St001059 Permutations ⟶ ℤ
Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation.
St001061 Permutations ⟶ ℤ
The number of indices that are both descents and recoils of a permutation.
St001074 Permutations ⟶ ℤ
The number of inversions of the cyclic embedding of a permutation.
St001076 Permutations ⟶ ℤ
The minimal length of a factorization of a permutation into transpositions that a....
St001077 Permutations ⟶ ℤ
The minimal length of a factorization of a permutation into star transpositions.
St001078 Permutations ⟶ ℤ
The minimal number of occurrences of (12) in a factorization of a permutation int....
St001079 Permutations ⟶ ℤ
The minimal length of a factorization of a permutation using the permutations (12....
St001080 Permutations ⟶ ℤ
The minimal length of a factorization of a permutation using the transposition (1....
St001081 Permutations ⟶ ℤ
The number of minimal length factorizations of a permutation into star transposit....
St001082 Permutations ⟶ ℤ
The number of boxed occurrences of 123 in a permutation.
St001083 Permutations ⟶ ℤ
The number of boxed occurrences of 132 in a permutation.
St001084 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |1-23 in a permutation.
St001085 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |21-3 in a permutation.
St001086 Permutations ⟶ ℤ
The number of occurrences of the consecutive pattern 132 in a permutation.
St001087 Permutations ⟶ ℤ
The number of occurrences of the vincular pattern |12-3 in a permutation.
St001090 Permutations ⟶ ℤ
The number of pop-stack-sorts needed to sort a permutation.
St001096 Permutations ⟶ ℤ
The size of the overlap set of a permutation.
St001114 Permutations ⟶ ℤ
The number of odd descents of a permutation.
St001115 Permutations ⟶ ℤ
The number of even descents of a permutation.
St001130 Permutations ⟶ ℤ
The number of two successive successions in a permutation.
St001160 Permutations ⟶ ℤ
The number of proper blocks (or intervals) of a permutations.
St001162 Permutations ⟶ ℤ
The minimum jump of a permutation.
St001168 Permutations ⟶ ℤ
The vector space dimension of the tilting module corresponding to the permutation....
St001171 Permutations ⟶ ℤ
The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corre....
St001174 Permutations ⟶ ℤ
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module correspo....
St001207 Permutations ⟶ ℤ
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding ....
St001208 Permutations ⟶ ℤ
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting m....
   Posets (51 statistics) # graph like objects
St000068 Posets ⟶ ℤ
The number of minimal elements in a poset.
St000069 Posets ⟶ ℤ
The number of maximal elements of a poset.
St000070 Posets ⟶ ℤ
The number of antichains in a poset.
St000071 Posets ⟶ ℤ
The number of maximal chains in a poset.
St000080 Posets ⟶ ℤ
The rank of the poset.
St000100 Posets ⟶ ℤ
The number of linear extensions of a poset.
St000104 Posets ⟶ ℤ
The number of facets in the order polytope of this poset.
St000151 Posets ⟶ ℤ
The number of facets in the chain polytope of the poset.
St000180 Posets ⟶ ℤ
The number of chains of a poset.
St000181 Posets ⟶ ℤ
The number of connected components of the Hasse diagram for the poset.
St000189 Posets ⟶ ℤ
The number of elements in the poset.
St000281 Posets ⟶ ℤ
The size of the preimage of the map 'to poset' from Binary trees to Posets.
St000282 Posets ⟶ ℤ
The size of the preimage of the map 'to poset' from Ordered trees to Posets.
St000298 Posets ⟶ ℤ
The order dimension or Dushnik-Miller dimension of a poset.
St000307 Posets ⟶ ℤ
The number of rowmotion orbits of a poset.
St000327 Posets ⟶ ℤ
The number of cover relations in a poset.
St000524 Posets ⟶ ℤ
The number of posets with the same order polynomial.
St000525 Posets ⟶ ℤ
The number of posets with the same zeta polynomial.
St000526 Posets ⟶ ℤ
The number of posets with combinatorially isomorphic order polytopes.
St000527 Posets ⟶ ℤ
The width of the poset.
St000528 Posets ⟶ ℤ
The height of a poset.
St000550 Posets ⟶ ℤ
The number of modular elements of a lattice.
St000551 Posets ⟶ ℤ
The number of left modular elements of a lattice.
St000632 Posets ⟶ ℤ
The jump number of the poset.
St000633 Posets ⟶ ℤ
The size of the automorphism group of a poset.
St000634 Posets ⟶ ℤ
The number of endomorphisms of a poset.
St000635 Posets ⟶ ℤ
The number of strictly order preserving maps of a poset into itself.
St000639 Posets ⟶ ℤ
The number of relations in a poset.
St000640 Posets ⟶ ℤ
The rank of the largest boolean interval in a poset.
St000641 Posets ⟶ ℤ
The number of non-empty boolean intervals in a poset.
St000642 Posets ⟶ ℤ
The size of the smallest orbit of antichains under Panyushev complementation.
St000643 Posets ⟶ ℤ
The size of the largest orbit of antichains under Panyushev complementation.
St000656 Posets ⟶ ℤ
The number of cuts of a poset.
St000680 Posets ⟶ ℤ
The Grundy value for Hackendot on posets.
St000717 Posets ⟶ ℤ
The number of ordinal summands of a poset.
St000845 Posets ⟶ ℤ
The maximal number of elements covered by an element in a poset.
St000846 Posets ⟶ ℤ
The maximal number of elements covering an element of a poset.
St000848 Posets ⟶ ℤ
The balance constant multiplied with the number of linear extensions of a poset.
St000849 Posets ⟶ ℤ
The number of 1/3-balanced pairs in a poset.
St000850 Posets ⟶ ℤ
The number of 1/2-balanced pairs in a poset.
St000906 Posets ⟶ ℤ
The length of the shortest maximal chain in a poset.
St000907 Posets ⟶ ℤ
The number of maximal antichains of minimal length in a poset.
St000908 Posets ⟶ ℤ
The length of the shortest maximal antichain in a poset.
St000909 Posets ⟶ ℤ
The number of maximal chains of maximal size in a poset.
St000910 Posets ⟶ ℤ
The number of maximal chains of minimal length in a poset.
St000911 Posets ⟶ ℤ
The number of maximal antichains of maximal size in a poset.
St000912 Posets ⟶ ℤ
The number of maximal antichains in a poset.
St000914 Posets ⟶ ℤ
The sum of the values of the Möbius function of a poset.
St001095 Posets ⟶ ℤ
The number of non-isomorphic posets with precisely one further covering relation.....
St001105 Posets ⟶ ℤ
The number of greedy linear extensions of a poset.
St001106 Posets ⟶ ℤ
The number of supergreedy linear extensions of a poset.
   Semistandard tableaux (11 statistics) # tableau like objects
St000101 Semistandard tableaux ⟶ ℤ
The cocharge of a semistandard tableau.
St000102 Semistandard tableaux ⟶ ℤ
The charge of a semistandard tableau.
St000103 Semistandard tableaux ⟶ ℤ
The sum of the entries of a semistandard tableau.
St000112 Semistandard tableaux ⟶ ℤ
The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is ....
St000116 Semistandard tableaux ⟶ ℤ
The major index of a semistandard tableau obtained by standardizing.
St000170 Semistandard tableaux ⟶ ℤ
The trace of a semistandard tableau.
St000173 Semistandard tableaux ⟶ ℤ
The segment statistic of a semistandard tableau.
St000174 Semistandard tableaux ⟶ ℤ
The flush statistic of a semistandard tableau.
St000736 Semistandard tableaux ⟶ ℤ
The last entry in the first row of a semistandard tableau.
St000737 Semistandard tableaux ⟶ ℤ
The last entry on the main diagonal of a semistandard tableau.
St000739 Semistandard tableaux ⟶ ℤ
The first entry in the last row of a semistandard tableau.
   Set partitions (101 statistics) # partition like objects
St000105 Set partitions ⟶ ℤ
The number of blocks in the set partition.
St000163 Set partitions ⟶ ℤ
The size of the orbit of the set partition under rotation.
St000211 Set partitions ⟶ ℤ
The rank of the set partition.
St000229 Set partitions ⟶ ℤ
Sum of the difference between the maximal and the minimal elements of the blocks ....
St000230 Set partitions ⟶ ℤ
Sum of the minimal elements of the blocks of a set partition.
St000231 Set partitions ⟶ ℤ
Sum of the maximal elements of the blocks of a set partition.
St000232 Set partitions ⟶ ℤ
The number of crossings of a set partition.
St000233 Set partitions ⟶ ℤ
The number of nestings of a set partition.
St000247 Set partitions ⟶ ℤ
The number of singleton blocks of a set partition.
St000248 Set partitions ⟶ ℤ
The number of anti-singletons of a set partition.
St000249 Set partitions ⟶ ℤ
The number of singletons (St000247The number of singleton blocks of a set partition.) plus the number of antisingletons (St000248The number of anti-singletons of a set partition.) ....
St000250 Set partitions ⟶ ℤ
The number of blocks (St000105The number of blocks in the set partition.) plus the number of antisingletons (St000248The number of anti-singletons of a set partition.) of a....
St000251 Set partitions ⟶ ℤ
The number of nonsingleton blocks of a set partition.
St000253 Set partitions ⟶ ℤ
The crossing number of a set partition.
St000254 Set partitions ⟶ ℤ
The nesting number of a set partition.
St000490 Set partitions ⟶ ℤ
The intertwining number of a set partition.
St000491 Set partitions ⟶ ℤ
The number of inversions of a set partition.
St000492 Set partitions ⟶ ℤ
The rob statistic of a set partition.
St000493 Set partitions ⟶ ℤ
The los statistic of a set partition.
St000496 Set partitions ⟶ ℤ
The rcs statistic of a set partition.
St000497 Set partitions ⟶ ℤ
The lcb statistic of a set partition.
St000498 Set partitions ⟶ ℤ
The lcs statistic of a set partition.
St000499 Set partitions ⟶ ℤ
The rcb statistic of a set partition.
St000502 Set partitions ⟶ ℤ
The number of successions of a set partitions.
St000503 Set partitions ⟶ ℤ
The maximal difference between two elements in a common block.
St000504 Set partitions ⟶ ℤ
The cardinality of the first block of a set partition.
St000505 Set partitions ⟶ ℤ
The biggest entry in the block containing the 1.
St000554 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,2},{3}} in a set partition.
St000555 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} in a set partition.
St000556 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} in a set partition.
St000557 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} in a set partition.
St000558 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,2}} in a set partition.
St000559 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.
St000560 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.
St000561 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,2,3}} in a set partition.
St000562 Set partitions ⟶ ℤ
The number of internal points of a set partition.
St000563 Set partitions ⟶ ℤ
The number of overlapping pairs of blocks of a set partition.
St000564 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} in a set partition.
St000565 Set partitions ⟶ ℤ
The major index of a set partition.
St000572 Set partitions ⟶ ℤ
The dimension exponent of a set partition.
St000573 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 ....
St000574 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a ....
St000575 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element ....
St000576 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a ....
St000577 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.....
St000578 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton.
St000579 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.....
St000580 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is m....
St000581 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is ma....
St000582 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is ma....
St000583 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 a....
St000584 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is m....
St000585 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) a....
St000586 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.
St000587 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal.
St000588 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 i....
St000589 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) a....
St000590 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is ma....
St000591 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal.
St000592 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal.
St000593 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal.
St000594 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3....
St000595 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal.
St000596 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is m....
St000597 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) a....
St000598 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is....
St000599 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive....
St000600 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) a....
St000601 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3....
St000602 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal.
St000603 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal.
St000604 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is m....
St000605 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) a....
St000606 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3....
St000607 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is ma....
St000608 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 i....
St000609 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
St000610 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal.
St000611 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal.
St000612 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) a....
St000613 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is ma....
St000614 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is ma....
St000615 Set partitions ⟶ ℤ
The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal.
St000695 Set partitions ⟶ ℤ
The number of blocks in the first part of the atomic decomposition of a set parti....
St000728 Set partitions ⟶ ℤ
The dimension of a set partition.
St000729 Set partitions ⟶ ℤ
The minimal arc length of a set partition.
St000730 Set partitions ⟶ ℤ
The maximal arc length of a set partition.
St000747 Set partitions ⟶ ℤ
A variant of the major index of a set partition.
St000748 Set partitions ⟶ ℤ
The major index of the permutation obtained by flattening the set partition.
St000793 Set partitions ⟶ ℤ
The length of the longest partition in the vacillating tableau corresponding to a....
St000823 Set partitions ⟶ ℤ
The number of unsplittable factors of the set partition.
St000839 Set partitions ⟶ ℤ
The largest opener of a set partition.
St000925 Set partitions ⟶ ℤ
The number of topologically connected components of a set partition.
St000971 Set partitions ⟶ ℤ
The smallest closer of a set partition.
St001050 Set partitions ⟶ ℤ
The number of terminal closers of a set partition.
St001051 Set partitions ⟶ ℤ
The depth of the label 1 in the decreasing labelled unordered tree associated wit....
St001062 Set partitions ⟶ ℤ
The maximal size of a block of a set partition.
St001075 Set partitions ⟶ ℤ
The minimal size of a block of a set partition.
St001094 Set partitions ⟶ ℤ
The depth index of a set partition.
St001151 Set partitions ⟶ ℤ
The number of blocks with odd minimum.
St001153 Set partitions ⟶ ℤ
The number of blocks with even minimum in a set partition.
   Standard tableaux (20 statistics) # tableau like objects
St000009 Standard tableaux ⟶ ℤ
The charge of a standard tableau.
St000016 Standard tableaux ⟶ ℤ
The number of attacking pairs of a standard tableau.
St000017 Standard tableaux ⟶ ℤ
The number of inversions of a standard tableau.
St000057 Standard tableaux ⟶ ℤ
The Shynar inversion number of a standard tableau.
St000059 Standard tableaux ⟶ ℤ
The inversion number of a standard tableau as defined by Haglund and Stevens.
St000075 Standard tableaux ⟶ ℤ
The orbit size of a standard tableau under promotion.
St000157 Standard tableaux ⟶ ℤ
The number of descents of a standard tableau.
St000169 Standard tableaux ⟶ ℤ
The cocharge of a standard tableau.
St000330 Standard tableaux ⟶ ℤ
The (standard) major index of a standard tableau.
St000336 Standard tableaux ⟶ ℤ
The leg major index of a standard tableau.
St000507 Standard tableaux ⟶ ℤ
The number of ascents of a standard tableau.
St000508 Standard tableaux ⟶ ℤ
Eigenvalues of the random-to-random operator acting on a simple module.
St000693 Standard tableaux ⟶ ℤ
The modular (standard) major index of a standard tableau.
St000733 Standard tableaux ⟶ ℤ
The row containing the largest entry of a standard tableau.
St000734 Standard tableaux ⟶ ℤ
The last entry in the first row of a standard tableau.
St000735 Standard tableaux ⟶ ℤ
The last entry on the main diagonal of a standard tableau.
St000738 Standard tableaux ⟶ ℤ
The first entry in the last row of a standard tableau.
St000743 Standard tableaux ⟶ ℤ
The number of entries in a standard Young tableau such that the next integer is a....
St000744 Standard tableaux ⟶ ℤ
The length of the path to the largest entry in a standard Young tableau.
St000745 Standard tableaux ⟶ ℤ
The index of the last row whose first entry is the row number in a standard Young....