- FindStat

Your data matches 657 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000047
(load all 2 compositions to match this statistic)

Mapping

St000047: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 1
[2,1] => 2
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 2
[1,3] => 1
[2,1,1] => 3
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 2
[1,1,3] => 1
[1,2,1,1] => 3
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 2
[1,1,1,3] => 1
[1,1,2,1,1] => 3
[1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => 1
[1,1,1,1,2,1] => 2
[1,1,1,1,3] => 1
[1,1,1,2,1,1] => 3

Description

The number of standard immaculate tableaux of a given shape. See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.

Matching statistic: St000766
(load all 4 compositions to match this statistic)

Mapping

St000766: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 = 1 - 1
[1,1] => 0 = 1 - 1
[2] => 0 = 1 - 1
[1,1,1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[3] => 0 = 1 - 1
[1,1,1,1] => 0 = 1 - 1
[1,1,2] => 0 = 1 - 1
[1,2,1] => 1 = 2 - 1
[1,3] => 0 = 1 - 1
[2,1,1] => 2 = 3 - 1
[1,1,1,1,1] => 0 = 1 - 1
[1,1,1,2] => 0 = 1 - 1
[1,1,2,1] => 1 = 2 - 1
[1,1,3] => 0 = 1 - 1
[1,2,1,1] => 2 = 3 - 1
[1,1,1,1,1,1] => 0 = 1 - 1
[1,1,1,1,2] => 0 = 1 - 1
[1,1,1,2,1] => 1 = 2 - 1
[1,1,1,3] => 0 = 1 - 1
[1,1,2,1,1] => 2 = 3 - 1
[1,1,1,1,1,1,1] => 0 = 1 - 1
[1,1,1,1,1,2] => 0 = 1 - 1
[1,1,1,1,2,1] => 1 = 2 - 1
[1,1,1,1,3] => 0 = 1 - 1
[1,1,1,2,1,1] => 2 = 3 - 1

Description

The number of inversions of an integer composition. This is the number of pairs

$(i,j)$ such that

$i < j$ and

$c_i > c_j$ .

Matching statistic: St001313
(load all 6 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 => 1
[1,1] => 11 => 1
[2] => 10 => 1
[1,1,1] => 111 => 1
[1,2] => 110 => 1
[2,1] => 101 => 2
[3] => 100 => 1
[1,1,1,1] => 1111 => 1
[1,1,2] => 1110 => 1
[1,2,1] => 1101 => 2
[1,3] => 1100 => 1
[2,1,1] => 1011 => 3
[1,1,1,1,1] => 11111 => 1
[1,1,1,2] => 11110 => 1
[1,1,2,1] => 11101 => 2
[1,1,3] => 11100 => 1
[1,2,1,1] => 11011 => 3
[1,1,1,1,1,1] => 111111 => 1
[1,1,1,1,2] => 111110 => 1
[1,1,1,2,1] => 111101 => 2
[1,1,1,3] => 111100 => 1
[1,1,2,1,1] => 111011 => 3
[1,1,1,1,1,1,1] => 1111111 => 1
[1,1,1,1,1,2] => 1111110 => 1
[1,1,1,1,2,1] => 1111101 => 2
[1,1,1,1,3] => 1111100 => 1
[1,1,1,2,1,1] => 1111011 => 3

Description

The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating

$1$ 's as steps

$(1,0)$ and

$0$ 's as steps

$(0,1)$ . Given a binary word

$w$ , this statistic counts the number of lattice paths from the origin to the same endpoint as

$w$ that stay weakly above

$w$ . See [[St001312]] for this statistic on compositions treated as bounce paths.

Matching statistic: St000682
(load all 6 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
St000682: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 => 0 = 1 - 1
[1,1] => 11 => 0 = 1 - 1
[2] => 10 => 0 = 1 - 1
[1,1,1] => 111 => 0 = 1 - 1
[1,2] => 110 => 0 = 1 - 1
[2,1] => 101 => 1 = 2 - 1
[3] => 100 => 0 = 1 - 1
[1,1,1,1] => 1111 => 0 = 1 - 1
[1,1,2] => 1110 => 0 = 1 - 1
[1,2,1] => 1101 => 1 = 2 - 1
[1,3] => 1100 => 0 = 1 - 1
[2,1,1] => 1011 => 2 = 3 - 1
[1,1,1,1,1] => 11111 => 0 = 1 - 1
[1,1,1,2] => 11110 => 0 = 1 - 1
[1,1,2,1] => 11101 => 1 = 2 - 1
[1,1,3] => 11100 => 0 = 1 - 1
[1,2,1,1] => 11011 => 2 = 3 - 1
[1,1,1,1,1,1] => 111111 => 0 = 1 - 1
[1,1,1,1,2] => 111110 => 0 = 1 - 1
[1,1,1,2,1] => 111101 => 1 = 2 - 1
[1,1,1,3] => 111100 => 0 = 1 - 1
[1,1,2,1,1] => 111011 => 2 = 3 - 1
[1,1,1,1,1,1,1] => 1111111 => 0 = 1 - 1
[1,1,1,1,1,2] => 1111110 => 0 = 1 - 1
[1,1,1,1,2,1] => 1111101 => 1 = 2 - 1
[1,1,1,1,3] => 1111100 => 0 = 1 - 1
[1,1,1,2,1,1] => 1111011 => 2 = 3 - 1

Description

The Grundy value of Welter's game on a binary word. Two players take turns moving a

$1$ to the left. The loosing positions are the words

$1\dots 10\dots 0$ .

Matching statistic: St000769
(load all 2 compositions to match this statistic)

Mapping

Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
St000769: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 0 = 1 - 1
[1,1] => [1,1] => 0 = 1 - 1
[2] => [2] => 0 = 1 - 1
[1,1,1] => [1,1,1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 1 = 2 - 1
[3] => [3] => 0 = 1 - 1
[1,1,1,1] => [1,1,1,1] => 0 = 1 - 1
[1,1,2] => [1,1,2] => 0 = 1 - 1
[1,2,1] => [2,1,1] => 1 = 2 - 1
[1,3] => [1,3] => 0 = 1 - 1
[2,1,1] => [1,2,1] => 2 = 3 - 1
[1,1,1,1,1] => [1,1,1,1,1] => 0 = 1 - 1
[1,1,1,2] => [1,1,1,2] => 0 = 1 - 1
[1,1,2,1] => [2,1,1,1] => 1 = 2 - 1
[1,1,3] => [1,1,3] => 0 = 1 - 1
[1,2,1,1] => [1,2,1,1] => 2 = 3 - 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 0 = 1 - 1
[1,1,1,1,2] => [1,1,1,1,2] => 0 = 1 - 1
[1,1,1,2,1] => [2,1,1,1,1] => 1 = 2 - 1
[1,1,1,3] => [1,1,1,3] => 0 = 1 - 1
[1,1,2,1,1] => [1,2,1,1,1] => 2 = 3 - 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0 = 1 - 1
[1,1,1,1,1,2] => [1,1,1,1,1,2] => 0 = 1 - 1
[1,1,1,1,2,1] => [2,1,1,1,1,1] => 1 = 2 - 1
[1,1,1,1,3] => [1,1,1,1,3] => 0 = 1 - 1
[1,1,1,2,1,1] => [1,2,1,1,1,1] => 2 = 3 - 1

Description

The major index of a composition regarded as a word. This is the sum of the positions of the descents of the composition. For the statistic which interprets the composition as a descent set, see [[St000008]].

Matching statistic: St001689
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001689: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => 0 = 1 - 1
[1,1] => ([(0,1)],2)
 => 0 = 1 - 1
[2] => ([],2)
 => 0 = 1 - 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[1,2] => ([(1,2)],3)
 => 0 = 1 - 1
[2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[3] => ([],3)
 => 0 = 1 - 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,3] => ([(2,3)],4)
 => 0 = 1 - 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 2 - 1
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 3 - 1

Description

The number of celebrities in a graph. A celebrity is a vertex such that more than half of its neighbours have smaller degree.

Matching statistic: St000063
(load all 5 compositions to match this statistic)

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1],[]]
 => []
 => 1
[1,1] => [[1,1],[]]
 => []
 => 1
[2] => [[2],[]]
 => []
 => 1
[1,1,1] => [[1,1,1],[]]
 => []
 => 1
[1,2] => [[2,1],[]]
 => []
 => 1
[2,1] => [[2,2],[1]]
 => [1]
 => 2
[3] => [[3],[]]
 => []
 => 1
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 1
[1,1,2] => [[2,1,1],[]]
 => []
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 2
[1,3] => [[3,1],[]]
 => []
 => 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 2
[1,1,3] => [[3,1,1],[]]
 => []
 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 2
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 2
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 3

Description

The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition

$\lambda$ also counts cover-inclusive Dyck tilings of

$\lambda\setminus\mu$ , summed over all

$\mu$ , as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.

Matching statistic: St000108
(load all 5 compositions to match this statistic)

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1],[]]
 => []
 => 1
[1,1] => [[1,1],[]]
 => []
 => 1
[2] => [[2],[]]
 => []
 => 1
[1,1,1] => [[1,1,1],[]]
 => []
 => 1
[1,2] => [[2,1],[]]
 => []
 => 1
[2,1] => [[2,2],[1]]
 => [1]
 => 2
[3] => [[3],[]]
 => []
 => 1
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 1
[1,1,2] => [[2,1,1],[]]
 => []
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 2
[1,3] => [[3,1],[]]
 => []
 => 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 2
[1,1,3] => [[3,1,1],[]]
 => []
 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 2
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 2
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 3

Description

The number of partitions contained in the given partition.

Matching statistic: St000532
(load all 5 compositions to match this statistic)

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1],[]]
 => []
 => 1
[1,1] => [[1,1],[]]
 => []
 => 1
[2] => [[2],[]]
 => []
 => 1
[1,1,1] => [[1,1,1],[]]
 => []
 => 1
[1,2] => [[2,1],[]]
 => []
 => 1
[2,1] => [[2,2],[1]]
 => [1]
 => 2
[3] => [[3],[]]
 => []
 => 1
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 1
[1,1,2] => [[2,1,1],[]]
 => []
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 2
[1,3] => [[3,1],[]]
 => []
 => 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 2
[1,1,3] => [[3,1,1],[]]
 => []
 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 2
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 2
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 3

Description

The total number of rook placements on a Ferrers board.

Matching statistic: St001400
(load all 5 compositions to match this statistic)

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001400: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1],[]]
 => []
 => 1
[1,1] => [[1,1],[]]
 => []
 => 1
[2] => [[2],[]]
 => []
 => 1
[1,1,1] => [[1,1,1],[]]
 => []
 => 1
[1,2] => [[2,1],[]]
 => []
 => 1
[2,1] => [[2,2],[1]]
 => [1]
 => 2
[3] => [[3],[]]
 => []
 => 1
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 1
[1,1,2] => [[2,1,1],[]]
 => []
 => 1
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 2
[1,3] => [[3,1],[]]
 => []
 => 1
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 2
[1,1,3] => [[3,1,1],[]]
 => []
 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 2
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 3
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 2
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 3

Description

The total number of Littlewood-Richardson tableaux of given shape. This is the multiplicity of the Schur function

$s_\lambda$ in

$\sum_{\mu, \nu} s_\mu s_\nu$ , where the sum is over all partitions

$\mu$ and

$\nu$ .

The following 647 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000010The length of the partition. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000355The number of occurrences of the pattern 21-3. St000384The maximal part of the shifted composition of an integer partition. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. 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. St000867The sum of the hook lengths in the first row of an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001127The sum of the squares of the parts of a partition. St001175The size of a partition minus the hook length of the base cell. St001323The independence gap of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000088The row sums of the character table of the symmetric group. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. 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. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000413The number of ordered trees with the same underlying unordered tree. St000531The leading coefficient of the rook polynomial of an integer partition. St000820The number of compositions obtained by rotating the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000935The number of ordered refinements of an integer partition. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001670The connected partition number of a graph. St001725The harmonious chromatic number of a graph. 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. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001814The number of partitions interlacing the given partition. St001881The number of factors of a lattice as a Cartesian product of lattices. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000057The Shynar inversion number of a standard tableau. St000065The number of entries equal to -1 in an alternating sign matrix. St000142The number of even parts of a partition. St000147The largest part of an integer partition. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000204The number of internal nodes of a binary tree. St000272The treewidth of a graph. St000291The number of descents of a binary word. St000358The number of occurrences of the pattern 31-2. St000362The size of a minimal vertex cover of a graph. 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

$3$ . St000378The diagonal inversion number of an integer partition. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000496The rcs statistic of a set partition. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000648The number of 2-excedences of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000877The depth of the binary word interpreted as a path. St000921The number of internal inversions of a binary word. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001120The length of a longest path in a graph. St001172The number of 1-rises at odd height of a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001586The number of odd parts smaller than the largest even part in an integer partition. St001644The dimension of a graph. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001727The number of invisible inversions of a permutation. St001743The discrepancy of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001792The arboricity of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001962The proper pathwidth of a graph. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000747A variant of the major index of a set partition. St000100The number of linear extensions of a poset. St000255The number of reduced Kogan faces with the permutation as type. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000017The number of inversions of a standard tableau. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000463The number of admissible inversions of a permutation. St000491The number of inversions of a set partition. St000538The number of even inversions of a permutation. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. 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. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (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. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. 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. 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. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. 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. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000359The number of occurrences of the pattern 23-1. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000647The number of big descents of a permutation. St001377The major index minus the number of inversions of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000524The number of posets with the same order polynomial. St001352The number of internal nodes in the modular decomposition of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001642The Prague dimension of a graph. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000462The major index minus the number of excedences of a permutation. St001280The number of parts of an integer partition that are at least two. St001592The maximal number of simple paths between any two different vertices of a graph. St001596The number of two-by-two squares inside a skew partition. St001657The number of twos in an integer partition. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000678The number of up steps after the last double rise of a Dyck path. St001268The size of the largest ordinal summand in the poset. St001779The order of promotion on the set of linear extensions of a poset. St000225Difference between largest and smallest parts in a partition. St000377The dinv defect of an integer partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001397Number of pairs of incomparable elements in a finite poset. St001911A descent variant minus the number of inversions. St000217The number of occurrences of the pattern 312 in a permutation. 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. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000091The descent variation of a composition. St000143The largest repeated part of a partition. St000222The number of alignments in the permutation. St000317The cycle descent number of a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000849The number of 1/3-balanced pairs in a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000961The shifted major index of a permutation. 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. 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001220The width of a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000711The number of big exceedences of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000667The greatest common divisor of the parts of the partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001933The largest multiplicity of a part in an integer partition. St000145The Dyson rank of a partition. St001176The size of a partition minus its first part. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000455The second largest eigenvalue of a graph if it is integral. St000045The number of linear extensions of a binary tree. St000993The multiplicity of the largest part of an integer partition. St000219The number of occurrences of the pattern 231 in a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000422The energy of a graph, if it is integral. St000516The number of stretching pairs of a permutation. St001535The number of cyclic alignments of a permutation. St001438The number of missing boxes of a skew partition. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001435The number of missing boxes in the first row. St001705The number of occurrences of the pattern 2413 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000133The "bounce" of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001868The number of alignments of type NE of a signed permutation. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001811The Castelnuovo-Mumford regularity of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. 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. St000618The number of self-evacuating tableaux of given shape. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. St001432The order dimension of the partition. 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. 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. 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. 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. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St000944The 3-degree of an integer partition. 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. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001964The interval resolution global dimension of a poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001249Sum of the odd parts of a 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. 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. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St000014The number of parking functions supported by a Dyck path. St000144The pyramid weight of the Dyck path. St000294The number of distinct factors of a binary word. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000439The position of the first down step of a Dyck path. St000456The monochromatic index of a connected graph. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. 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. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000048The multinomial of the parts of a partition. St000117The number of centered tunnels of a Dyck path. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000288The number of ones in a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000392The length of the longest run of ones in a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000438The position of the last up step in a Dyck path. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000627The exponent of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. 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. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000876The number of factors in the Catalan decomposition of a binary word. 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. St000922The minimal number such that all substrings of this length are unique. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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

$g$ . 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. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001224Let X be the direct sum of all simple modules 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001481The minimal height of a peak of a Dyck path. St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001732The number of peaks visible from the left. St001809The index of the step at the first peak of maximal height in a Dyck path. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St001910The height of the middle non-run of a Dyck path. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000012The area of a Dyck path. 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. St000185The weighted size of a 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. St000295The length of the border of a binary word. 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. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. 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. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000869The sum of the hook lengths of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000984The number of boxes below precisely one peak. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of 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. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with 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. St001139The number of occurrences of hills of size 2 in a Dyck path. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001413Half the length of the longest even length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001484The number of singletons of an integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001524The degree of symmetry of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001697The shifted natural comajor index of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. 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). St001930The weak major index of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St001956The comajor index for set-valued two-row standard Young tableaux. St001128The exponens consonantiae of a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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. St000562The number of internal points of a set 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. St000181The number of connected components of the Hasse diagram for the poset. St001060The distinguishing index of a graph. St001545The second Elser number of a connected graph. St001890The maximum magnitude of the Möbius function of a poset. St000264The girth of a graph, which is not a tree. St001570The minimal number of edges to add to make a graph Hamiltonian. St000464The Schultz index of a connected graph. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001645The pebbling number of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000706The product of the factorials of the multiplicities of an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the 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. 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. St001568The smallest positive integer that does not appear twice in the partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000978The sum of the positions of double down-steps of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000977MacMahon's equal index of a Dyck path. St000981The length of the longest zigzag subpath. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. 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. St000693The modular (standard) major index of a standard tableau. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. 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. 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. 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. St001808The box weight or horizontal decoration of a Dyck path. 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. St001943The sum of the squares of the hook lengths of an integer partition. 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. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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. St001141The number of occurrences of hills of size 3 in a Dyck path. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. 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. St001651The Frankl number of a lattice. St000474Dyson's crank of a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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

$K[x]/(x^n)$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001556The number of inversions of the third entry of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.