Your data matches 686 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000256
(load all 17 compositions to match this statistic)
Mapping
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 1
[2,1]
 => 0
[1,1,1]
 => 0
[4]
 => 1
[3,1]
 => 1
[2,2]
 => 1
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 1
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000257
(load all 11 compositions to match this statistic)
Mapping
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 0
[1,1,1]
 => 1
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 1
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 1
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St001092
(load all 13 compositions to match this statistic)
Mapping
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 1
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 0
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St000386
(load all 115 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001037
(load all 73 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => 0
[1,1]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 0
[2,1]
 => [1,0,1,0,1,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000023
(load all 31 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
Description
The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].
Matching statistic: St000196
(load all 22 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [[.,.],.]
 => 0
[2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 0
[1,1]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 0
[2,1]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000353
(load all 33 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 0
Description
The number of inner valleys of a permutation. The number of valleys including the boundary is [[St000099]].
Matching statistic: St000390
(load all 19 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 0 => 0
[2]
 => 0 => 1 => 1
[1,1]
 => 11 => 00 => 0
[3]
 => 1 => 0 => 0
[2,1]
 => 01 => 10 => 1
[1,1,1]
 => 111 => 000 => 0
[4]
 => 0 => 1 => 1
[3,1]
 => 11 => 00 => 0
[2,2]
 => 00 => 11 => 1
[2,1,1]
 => 011 => 100 => 1
[1,1,1,1]
 => 1111 => 0000 => 0
[5]
 => 1 => 0 => 0
[4,1]
 => 01 => 10 => 1
[3,2]
 => 10 => 01 => 1
[3,1,1]
 => 111 => 000 => 0
[2,2,1]
 => 001 => 110 => 1
[2,1,1,1]
 => 0111 => 1000 => 1
[1,1,1,1,1]
 => 11111 => 00000 => 0
Description
The number of runs of ones in a binary word.
Matching statistic: St000624
(load all 33 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
Description
The normalized sum of the minimal distances to a greater element. Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is $$ \sum_{i=1}^n \min_d(\pi_{i-1-d}>\pi_i\text{ or }\pi_{i+1+d}>\pi_i) $$ A closely related statistic appears in [1]. The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently satisfies the functional equation $$ (x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x^3 = 0. $$
The following 676 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000779The tier of a permutation. St001115The number of even descents of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000201The number of leaf nodes in a binary tree. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000396The register function (or Horton-Strahler number) of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000862The number of parts of the shifted shape of a permutation. St000920The logarithmic height of a Dyck path. St000035The number of left outer peaks of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000241The number of cyclical small excedances. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000374The number of exclusive right-to-left minima of a permutation. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000538The number of even inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000628The balance of a binary word. St000632The jump number of the poset. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000665The number of rafts of a permutation. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000834The number of right outer peaks of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000836The number of descents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000873The aix statistic of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001114The number of odd descents of a permutation. St001153The number of blocks with even minimum in a set partition. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001394The genus of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001423The number of distinct cubes in a binary word. St001469The holeyness of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001712The number of natural descents of a standard Young tableau. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000007The number of saliances of the permutation. St000021The number of descents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000120The number of left tunnels of a Dyck path. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000451The length of the longest pattern of the form k 1 2. St000495The number of inversions of distance at most 2 of a permutation. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000568The hook number of a binary tree. St000630The length of the shortest palindromic decomposition of a binary word. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000679The pruning number of an ordered tree. St000701The protection number of a binary tree. St000733The row containing the largest entry of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000829The Ulam distance of a permutation to the identity permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000883The number of longest increasing subsequences of a permutation. St000884The number of isolated descents of a permutation. St000886The number of permutations with the same antidiagonal sums. St000903The number of different parts of an integer composition. St000919The number of maximal left branches of a binary tree. St000983The length of the longest alternating subword. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001151The number of blocks with odd minimum. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001512The minimum rank of a graph. St001571The Cartan determinant of the integer partition. St001642The Prague dimension of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000325The width of the tree associated to a permutation. St000397The Strahler number of a rooted tree. St000470The number of runs in a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001093The detour number of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000355The number of occurrences of the pattern 21-3. St001095The number of non-isomorphic posets with precisely one further covering relation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001948The number of augmented double ascents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000996The number of exclusive left-to-right maxima of a permutation. St001928The number of non-overlapping descents in a permutation. St001933The largest multiplicity of a part in an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001960The number of descents of a permutation minus one if its first entry is not one. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001729The number of visible descents of a permutation. St000252The number of nodes of degree 3 of a binary tree. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001549The number of restricted non-inversions between exceedances. St001587Half of the largest even part of an integer partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001487The number of inner corners of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001665The number of pure excedances of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000658The number of rises of length 2 of a Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000455The second largest eigenvalue of a graph if it is integral. St000618The number of self-evacuating tableaux of given shape. St001435The number of missing boxes in the first row. St001498The normalised height of a Nakayama algebra with magnitude 1. St000259The diameter of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000137The Grundy value of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000378The diagonal inversion number of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000547The number of even non-empty partial sums of an integer partition. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001484The number of singletons of an integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000148The number of odd parts of a partition. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000228The size of a partition. St000234The number of global ascents of a permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000253The crossing number of a set partition. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000409The number of pitchforks in a binary tree. St000445The number of rises of length 1 of a Dyck path. St000459The hook length of the base cell of a partition. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000475The number of parts equal to 1 in a partition. St000507The number of ascents of a standard tableau. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000650The number of 3-rises of a permutation. St000651The maximal size of a rise in a permutation. St000663The number of right floats of a permutation. St000676The number of odd rises of a Dyck path. St000730The maximal arc length of a set partition. St000766The number of inversions of an integer composition. St000784The maximum of the length and the largest part of the integer partition. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000867The sum of the hook lengths in the first row of an integer partition. St000868The aid statistic in the sense of Shareshian-Wachs. St000869The sum of the hook lengths of an integer partition. St000872The number of very big descents of a permutation. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000944The 3-degree of an integer partition. St000992The alternating sum of the parts of an integer partition. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001127The sum of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001375The pancake length of a permutation. St001388The number of non-attacking neighbors of a permutation. St001403The number of vertical separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001524The degree of symmetry of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St001557The number of inversions of the second entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001657The number of twos in an integer partition. St001673The degree of asymmetry of an integer composition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001697The shifted natural comajor index of a standard Young tableau. St001726The number of visible inversions of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000807The sum of the heights of the valleys of the associated bargraph. St000929The constant term of the character polynomial of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000805The number of peaks of the associated bargraph. St001569The maximal modular displacement of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000383The last part of an integer composition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000941The number of characters of the symmetric group whose value on the partition is even. St000442The maximal area to the right of an up step of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001139The number of occurrences of hills of size 2 in a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000219The number of occurrences of the pattern 231 in a permutation. St000260The radius of a connected graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000472The sum of the ascent bottoms of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000478Another weight of a partition according to Alladi. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000539The number of odd inversions of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000565The major index of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000677The standardized bi-alternating inversion number of a permutation. St000693The modular (standard) major index of a standard tableau. St000728The dimension of a set partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000833The comajor index of a permutation. St000837The number of ascents of distance 2 of a permutation. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000928The sum of the coefficients of the character polynomial of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001480The number of simple summands of the module J^2/J^3. St001541The Gini index of an integer partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001721The degree of a binary word. St001731The factorization defect of a permutation. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000003The number of standard Young tableaux of the partition. St000026The position of the first return of a Dyck path. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000075The orbit size of a standard tableau under promotion. St000088The row sums of the character table of the symmetric group. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000182The number of permutations whose cycle type is the given integer partition. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000377The dinv defect of an integer partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000644The number of graphs with given frequency partition. St000649The number of 3-excedences of a permutation. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000913The number of ways to refine the partition into singletons. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001161The major index north count of a Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001214The aft of an integer partition. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001252Half the sum of the even parts of a partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001313The number of Dyck paths above the lattice path given by a binary word. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001595The number of standard Young tableaux of the skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001162The minimum jump of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000456The monochromatic index of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001527The cyclic permutation representation number of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001128The exponens consonantiae of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001943The sum of the squares of the hook lengths of an integer partition. St000054The first entry of the permutation. St000045The number of linear extensions of a binary tree. St000091The descent variation of a composition. St000284The Plancherel distribution on integer partitions. St000365The number of double ascents of a permutation. St000562The number of internal points of a set partition. St000567The sum of the products of all pairs of parts. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001438The number of missing boxes of a skew partition. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000090The variation of a composition. St000491The number of inversions of a set partition. St000492The rob statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000089The absolute variation of a composition. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000522The number of 1-protected nodes of a rooted tree. St000839The largest opener of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000521The number of distinct subtrees of an ordered tree. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000509The diagonal index (content) of a partition. St000674The number of hills of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000939The number of characters of the symmetric group whose value on the partition is positive. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000993The multiplicity of the largest part of an integer partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000307The number of rowmotion orbits of a poset. St000717The number of ordinal summands of a poset.