Your data matches 680 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001091
(load all 3 compositions to match this statistic)
Mapping
St001091: 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]
 => 2
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 3
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 4
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St001571
(load all 6 compositions to match this statistic)
Mapping
St001571: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 0 + 1
[2]
 => 2 = 1 + 1
[1,1]
 => 1 = 0 + 1
[3]
 => 3 = 2 + 1
[2,1]
 => 1 = 0 + 1
[1,1,1]
 => 1 = 0 + 1
[4]
 => 4 = 3 + 1
[3,1]
 => 2 = 1 + 1
[2,2]
 => 2 = 1 + 1
[2,1,1]
 => 1 = 0 + 1
[1,1,1,1]
 => 1 = 0 + 1
[5]
 => 5 = 4 + 1
[4,1]
 => 3 = 2 + 1
[3,2]
 => 2 = 1 + 1
[3,1,1]
 => 2 = 1 + 1
[2,2,1]
 => 1 = 0 + 1
[2,1,1,1]
 => 1 = 0 + 1
[1,1,1,1,1]
 => 1 = 0 + 1
Description
The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St001933
(load all 4 compositions to match this statistic)
Mapping
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 0 + 1
[2]
 => 1 = 0 + 1
[1,1]
 => 2 = 1 + 1
[3]
 => 1 = 0 + 1
[2,1]
 => 1 = 0 + 1
[1,1,1]
 => 3 = 2 + 1
[4]
 => 1 = 0 + 1
[3,1]
 => 1 = 0 + 1
[2,2]
 => 2 = 1 + 1
[2,1,1]
 => 2 = 1 + 1
[1,1,1,1]
 => 4 = 3 + 1
[5]
 => 1 = 0 + 1
[4,1]
 => 1 = 0 + 1
[3,2]
 => 1 = 0 + 1
[3,1,1]
 => 2 = 1 + 1
[2,2,1]
 => 2 = 1 + 1
[2,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1]
 => 5 = 4 + 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000120
(load all 14 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,1]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
Description
The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St000392
(load all 9 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 1 = 0 + 1
[2]
 => 100 => 1 = 0 + 1
[1,1]
 => 110 => 2 = 1 + 1
[3]
 => 1000 => 1 = 0 + 1
[2,1]
 => 1010 => 1 = 0 + 1
[1,1,1]
 => 1110 => 3 = 2 + 1
[4]
 => 10000 => 1 = 0 + 1
[3,1]
 => 10010 => 1 = 0 + 1
[2,2]
 => 1100 => 2 = 1 + 1
[2,1,1]
 => 10110 => 2 = 1 + 1
[1,1,1,1]
 => 11110 => 4 = 3 + 1
[5]
 => 100000 => 1 = 0 + 1
[4,1]
 => 100010 => 1 = 0 + 1
[3,2]
 => 10100 => 1 = 0 + 1
[3,1,1]
 => 100110 => 2 = 1 + 1
[2,2,1]
 => 11010 => 2 = 1 + 1
[2,1,1,1]
 => 101110 => 3 = 2 + 1
[1,1,1,1,1]
 => 111110 => 5 = 4 + 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 10 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 1 = 0 + 1
[2]
 => 100 => 1 = 0 + 1
[1,1]
 => 110 => 2 = 1 + 1
[3]
 => 1000 => 1 = 0 + 1
[2,1]
 => 1010 => 1 = 0 + 1
[1,1,1]
 => 1110 => 3 = 2 + 1
[4]
 => 10000 => 1 = 0 + 1
[3,1]
 => 10010 => 1 = 0 + 1
[2,2]
 => 1100 => 2 = 1 + 1
[2,1,1]
 => 10110 => 2 = 1 + 1
[1,1,1,1]
 => 11110 => 4 = 3 + 1
[5]
 => 100000 => 1 = 0 + 1
[4,1]
 => 100010 => 1 = 0 + 1
[3,2]
 => 10100 => 1 = 0 + 1
[3,1,1]
 => 100110 => 2 = 1 + 1
[2,2,1]
 => 11010 => 2 = 1 + 1
[2,1,1,1]
 => 101110 => 3 = 2 + 1
[1,1,1,1,1]
 => 111110 => 5 = 4 + 1
Description
The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St001498
(load all 25 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,1]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000052
(load all 16 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 0
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000355
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000355: 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]
 => [1,3,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 4
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000356
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000356: 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] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[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] => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0
[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] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
[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] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 4
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
The following 670 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000358The number of occurrences of the pattern 31-2. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000876The number of factors in the Catalan decomposition of a binary word. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000982The length of the longest constant subword. St000996The number of exclusive left-to-right maxima of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001733The number of weak left to right maxima of a Dyck path. St000381The largest part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000983The length of the longest alternating subword. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000074The number of special entries. St000143The largest repeated part of a partition. St000204The number of internal nodes of a binary tree. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000288The number of ones in a binary word. St000295The length of the border of a binary word. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000354The number of recoils of a permutation. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000441The number of successions of a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (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. St000662The staircase size of the code of a permutation. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. 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. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000877The depth of the binary word interpreted as a path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001130The number of two successive successions in a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive 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. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001377The major index minus the number of inversions of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001489The maximum of the number of descents and the number of inverse descents. St001584The area statistic between a Dyck path and its bounce path. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001910The height of the middle non-run of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000053The number of valleys of the Dyck path. St000056The decomposition (or block) number of a permutation. St000153The number of adjacent cycles of a permutation. St000160The multiplicity of the smallest part of a partition. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000216The absolute length of a permutation. St000225Difference between largest and smallest parts in a partition. St000314The number of left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000325The width of the tree associated to a permutation. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000383The last part of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000470The number of runs in a permutation. St000475The number of parts equal to 1 in a partition. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000654The first descent of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000730The maximal arc length of a set partition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000809The reduced reflection length of the permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000956The maximal displacement of a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000991The number of right-to-left minima 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. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the 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. St001050The number of terminal closers of a set partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001220The width of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001399The distinguishing number of a poset. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001726The number of visible inversions of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001884The number of borders of a binary word. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000147The largest part of an integer partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001366The maximal multiplicity of a degree of a vertex of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St001654The monophonic hull number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000297The number of leading ones in a binary word. St001948The number of augmented double ascents of a permutation. St000675The number of centered multitunnels of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000740The last entry of a permutation. St000798The makl of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000989The number of final rises of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001513The number of nested exceedences of a permutation. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000051The size of the left subtree of a binary tree. 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. St000461The rix statistic 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. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. 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. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001684The reduced word complexity of a permutation. St000004The major index of a permutation. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000305The inverse major index of a permutation. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000025The number of initial rises of a Dyck path. St000738The first entry in the last row of a standard tableau. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000993The multiplicity of the largest part of an integer partition. St001438The number of missing boxes of a skew partition. St001868The number of alignments of type NE of a signed permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001176The size of a partition minus its first part. St000010The length of the partition. St000012The area of a Dyck path. St000142The number of even parts of a partition. St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000185The weighted size of a partition. St000228The size of a partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. 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. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000459The hook length of the base cell of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001127The sum of the squares of the parts of a partition. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001274The number of indecomposable injective modules with projective dimension equal to two. St001280The number of parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001657The number of twos in an integer partition. St001730The number of times the path corresponding to a binary word crosses the base line. St001867The number of alignments of type EN of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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 $eA$. St001570The minimal number of edges to add to make a graph Hamiltonian. St000406The number of occurrences of the pattern 3241 in a permutation. St000516The number of stretching pairs of a permutation. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000873The aix statistic of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001847The number of occurrences of the pattern 1432 in a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001864The number of excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001937The size of the center of a parking function. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000477The weight of a partition according to Alladi. St001435The number of missing boxes in the first row. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001487The number of inner corners of a skew partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000454The largest eigenvalue of a graph if it is integral. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. 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. St000941The number of characters of the symmetric group whose value on the partition is even. 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. 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. 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. St000083The number of left oriented leafs of a binary tree except the first one. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000353The number of inner valleys of a permutation. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000472The sum of the ascent bottoms of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000498The lcs statistic of a set partition. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000519The largest length of a factor maximising the subword complexity. St000538The number of even inversions of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. 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. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000646The number of big ascents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000677The standardized bi-alternating inversion number of a permutation. 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. St000693The modular (standard) major index of a standard tableau. St000728The dimension of a set partition. St000779The tier of a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000961The shifted major index of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001139The number of occurrences of hills of size 2 in a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001371The length of the longest Yamanouchi prefix of a binary word. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001388The number of non-attacking neighbors of a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001480The number of simple summands of the module J^2/J^3. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001524The degree of symmetry of a binary word. 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. St001587Half of the largest even part of an integer partition. St001721The degree of a binary word. St001731The factorization defect of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001330The hat guessing number of a graph. St000944The 3-degree of an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000145The Dyson rank 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000183The side length of the Durfee square of an integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000306The bounce count of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000547The number of even non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000661The number of rises of length 3 of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. 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. St000783The side length of the largest staircase partition fitting into a partition. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000921The number of internal inversions of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 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. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001092The number of distinct even 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. 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. 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. 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001423The number of distinct cubes in a binary word. St001484The number of singletons of an integer partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001556The number of inversions of the third entry of a permutation. 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. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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). St001955The number of natural descents for set-valued two row standard Young tableaux. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. 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. St000259The diameter of a connected graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001378The product of the cohook lengths of the integer partition. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. 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. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a 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. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000284The Plancherel distribution on integer partitions. St000460The hook length of the last cell along the main diagonal of an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a 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. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. 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. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000352The Elizalde-Pak rank of a permutation. St001645The pebbling number of a connected graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. 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. St000045The number of linear extensions of a binary tree. St000091The descent variation of a composition. St000219The number of occurrences of the pattern 231 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000562The number of internal points of a set partition. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. St001128The exponens consonantiae of a partition. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001568The smallest positive integer that does not appear twice in the partition. St000090The variation of a composition. St000492The rob statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St001151The number of blocks with odd minimum. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. 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. St001375The pancake length of a permutation. 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. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. 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. 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. St000420The number of Dyck paths that are weakly above a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000509The diagonal index (content) of a partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000744The length of the path to the largest entry in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000947The major index east count of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001520The number of strict 3-descents. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. 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. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001808The box weight or horizontal decoration of a Dyck path. 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. St001943The sum of the squares of the hook lengths of an integer partition. St000632The jump number of the poset. 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 $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. 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.