**Identifier**

**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

**Dyck paths**(148 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 isimple 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.

**Finite Cartan types**(35 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.

**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 VerdiÃ¨re 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**(146 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.

**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**(9 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 cosum of 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.....

**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**(290 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 SteingrÃmsson'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.

**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 MÃ¶bius 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

St000250
Set partitions ⟶ ℤ

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....