Your data matches 469 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000228
(load all 8 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([],6)
 => []
 => 0
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [2]
 => 2
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 3
([],7)
 => []
 => 0
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [2]
 => 2
([(3,6),(4,5)],7)
 => [1,1]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000384
(load all 6 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([],6)
 => []
 => 0
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [2]
 => 2
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 3
([],7)
 => []
 => 0
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [2]
 => 2
([(3,6),(4,5)],7)
 => [1,1]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 3
Description
The maximal part of the shifted composition of an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part. The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$. See also [[St000380]].
Matching statistic: St000459
(load all 8 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([],6)
 => []
 => 0
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [2]
 => 2
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 3
([],7)
 => []
 => 0
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [2]
 => 2
([(3,6),(4,5)],7)
 => [1,1]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 3
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000784
(load all 6 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([],6)
 => []
 => 0
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [2]
 => 2
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 3
([],7)
 => []
 => 0
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [2]
 => 2
([(3,6),(4,5)],7)
 => [1,1]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 3
Description
The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].
Matching statistic: St001621
Mapping
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],1)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],1)
 => 0
([(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([],4)
 => ([],1)
 => 0
([(2,3)],4)
 => ([(0,1)],2)
 => 1
([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([],5)
 => ([],1)
 => 0
([(3,4)],5)
 => ([(0,1)],2)
 => 1
([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([],6)
 => ([],1)
 => 0
([(4,5)],6)
 => ([(0,1)],2)
 => 1
([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([],7)
 => ([],1)
 => 0
([(5,6)],7)
 => ([(0,1)],2)
 => 1
([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
Description
The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001622
Mapping
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => 0
([],2)
 => ([],1)
 => 0
([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],1)
 => 0
([(1,2)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([],4)
 => ([],1)
 => 0
([(2,3)],4)
 => ([(0,1)],2)
 => 1
([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([],5)
 => ([],1)
 => 0
([(3,4)],5)
 => ([(0,1)],2)
 => 1
([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([],6)
 => ([],1)
 => 0
([(4,5)],6)
 => ([(0,1)],2)
 => 1
([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
([],7)
 => ([],1)
 => 0
([(5,6)],7)
 => ([(0,1)],2)
 => 1
([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
Description
The number of join-irreducible elements of a lattice. An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St000063
(load all 6 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000063: 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
([(0,1)],2)
 => [1]
 => 2 = 1 + 1
([],3)
 => []
 => 1 = 0 + 1
([(1,2)],3)
 => [1]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2]
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 4 = 3 + 1
([],4)
 => []
 => 1 = 0 + 1
([(2,3)],4)
 => [1]
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 4 = 3 + 1
([],5)
 => []
 => 1 = 0 + 1
([(3,4)],5)
 => [1]
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [2]
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 4 = 3 + 1
([],6)
 => []
 => 1 = 0 + 1
([(4,5)],6)
 => [1]
 => 2 = 1 + 1
([(3,5),(4,5)],6)
 => [2]
 => 3 = 2 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 4 = 3 + 1
([],7)
 => []
 => 1 = 0 + 1
([(5,6)],7)
 => [1]
 => 2 = 1 + 1
([(4,6),(5,6)],7)
 => [2]
 => 3 = 2 + 1
([(3,6),(4,5)],7)
 => [1,1]
 => 3 = 2 + 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 4 = 3 + 1
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 6 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000108: 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
([(0,1)],2)
 => [1]
 => 2 = 1 + 1
([],3)
 => []
 => 1 = 0 + 1
([(1,2)],3)
 => [1]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2]
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 4 = 3 + 1
([],4)
 => []
 => 1 = 0 + 1
([(2,3)],4)
 => [1]
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 4 = 3 + 1
([],5)
 => []
 => 1 = 0 + 1
([(3,4)],5)
 => [1]
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [2]
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 4 = 3 + 1
([],6)
 => []
 => 1 = 0 + 1
([(4,5)],6)
 => [1]
 => 2 = 1 + 1
([(3,5),(4,5)],6)
 => [2]
 => 3 = 2 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 4 = 3 + 1
([],7)
 => []
 => 1 = 0 + 1
([(5,6)],7)
 => [1]
 => 2 = 1 + 1
([(4,6),(5,6)],7)
 => [2]
 => 3 = 2 + 1
([(3,6),(4,5)],7)
 => [1,1]
 => 3 = 2 + 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 4 = 3 + 1
Description
The number of partitions contained in the given partition.
Matching statistic: St000532
(load all 6 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000532: 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
([(0,1)],2)
 => [1]
 => 2 = 1 + 1
([],3)
 => []
 => 1 = 0 + 1
([(1,2)],3)
 => [1]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2]
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 4 = 3 + 1
([],4)
 => []
 => 1 = 0 + 1
([(2,3)],4)
 => [1]
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 4 = 3 + 1
([],5)
 => []
 => 1 = 0 + 1
([(3,4)],5)
 => [1]
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [2]
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 4 = 3 + 1
([],6)
 => []
 => 1 = 0 + 1
([(4,5)],6)
 => [1]
 => 2 = 1 + 1
([(3,5),(4,5)],6)
 => [2]
 => 3 = 2 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 4 = 3 + 1
([],7)
 => []
 => 1 = 0 + 1
([(5,6)],7)
 => [1]
 => 2 = 1 + 1
([(4,6),(5,6)],7)
 => [2]
 => 3 = 2 + 1
([(3,6),(4,5)],7)
 => [1,1]
 => 3 = 2 + 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 4 = 3 + 1
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St001400
(load all 6 compositions to match this statistic)
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001400: 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
([(0,1)],2)
 => [1]
 => 2 = 1 + 1
([],3)
 => []
 => 1 = 0 + 1
([(1,2)],3)
 => [1]
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2]
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 4 = 3 + 1
([],4)
 => []
 => 1 = 0 + 1
([(2,3)],4)
 => [1]
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [1,1]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 4 = 3 + 1
([],5)
 => []
 => 1 = 0 + 1
([(3,4)],5)
 => [1]
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [2]
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => [1,1]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 4 = 3 + 1
([],6)
 => []
 => 1 = 0 + 1
([(4,5)],6)
 => [1]
 => 2 = 1 + 1
([(3,5),(4,5)],6)
 => [2]
 => 3 = 2 + 1
([(2,5),(3,4)],6)
 => [1,1]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 4 = 3 + 1
([],7)
 => []
 => 1 = 0 + 1
([(5,6)],7)
 => [1]
 => 2 = 1 + 1
([(4,6),(5,6)],7)
 => [2]
 => 3 = 2 + 1
([(3,6),(4,5)],7)
 => [1,1]
 => 3 = 2 + 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 4 = 3 + 1
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 459 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000171The degree of the graph. St000185The weighted size of a partition. St000479The Ramsey number of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000741The Colin de Verdière graph invariant. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001034The area of the parallelogram polyomino associated with the Dyck path. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001391The disjunction number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000025The number of initial rises of a Dyck path. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000169The cocharge of a standard tableau. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000676The number of odd rises of a Dyck path. St001697The shifted natural comajor index of a standard Young tableau. St000081The number of edges of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000984The number of boxes below precisely one peak. St000454The largest eigenvalue of a graph if it is integral. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000086The number of subgraphs. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St000448The number of pairs of vertices of a graph with distance 2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001959The product of the heights of the peaks of a Dyck path. St000087The number of induced subgraphs. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000926The clique-coclique number of a graph. St001249Sum of the odd parts of a partition. St001342The number of vertices in the center of a graph. St001360The number of covering relations in Young's lattice below a partition. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000778The metric dimension of a graph. St001345The Hamming dimension of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001949The rigidity index of a graph. St000048The multinomial of the parts of a partition. St000088The row sums of the character table of the symmetric group. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000258The burning number of a graph. St000293The number of inversions of a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000395The sum of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000667The greatest common divisor of the parts of the partition. 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. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000910The number of maximal chains of minimal length in a poset. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St000935The number of ordered refinements of an integer partition. St000982The length of the longest constant subword. St000992The alternating sum of the parts of an integer partition. St001007Number of simple modules with projective dimension 1 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. 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. St001055The Grundy value for the game of removing cells of a row in an integer partition. 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. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. 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. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001389The number of partitions of the same length below the given integer partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001486The number of corners of the ribbon associated with an integer composition. 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. 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. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001672The restrained domination number of a graph. St001757The number of orbits of toric promotion on a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000014The number of parking functions supported by a Dyck path. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. 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. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000420The number of Dyck paths that are weakly above a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. 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. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions 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. 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in 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. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second 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 St001267The length of the Lyndon factorization of the binary word. St001340The cardinality of a minimal non-edge isolating set of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001437The flex of a binary word. St001480The number of simple summands of the module J^2/J^3. 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. 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. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001012Number of simple modules with projective dimension at most 2 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000290The major index of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000678The number of up steps after the last double rise of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000946The sum of the skew hook positions in a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules 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. St001161The major index north count of a Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to 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 Dyck path as follows: 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001485The modular major index of a binary word. St001500The global dimension of magnitude 1 Nakayama algebras. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001614The cyclic permutation representation number of a skew partition. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001956The comajor index for set-valued two-row standard Young tableaux. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000306The bounce count of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000369The dinv deficit of a Dyck path. St000439The position of the first down step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000674The number of hills of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000792The Grundy value for the game of ruler on a binary word. St000874The position of the last double rise in a Dyck path. St000876The number of factors in the Catalan decomposition of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. 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 $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000981The length of the longest zigzag subpath. 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. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two 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. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001584The area statistic between a Dyck path and its bounce path. 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). St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000456The monochromatic index of a connected graph. St001645The pebbling number of a connected graph. St001118The acyclic chromatic index of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001875The number of simple modules with projective dimension at most 1. St000379The number of Hamiltonian cycles in a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000934The 2-degree of an integer partition. St000997The even-odd crank of an integer partition. St000477The weight of a partition according to Alladi. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. 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. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000744The length of the path to the largest entry in a standard Young tableau. St000806The semiperimeter of the associated bargraph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. St001128The exponens consonantiae of a partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to 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. St000509The diagonal index (content) of a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000438The position of the last up step in a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000976The sum of the positions of double up-steps of a 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. 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. 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. St001141The number of occurrences of hills of size 3 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001378The product of the cohook lengths of the integer partition. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. 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. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition 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. 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. 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. St001383The BG-rank of an integer partition. St001432The order dimension of the 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. 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. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on 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. St001610The number of coloured endofunctions 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001877Number of indecomposable injective modules with projective dimension 2. 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. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a 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. 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. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000474Dyson's crank of a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001330The hat guessing number of a graph. St001651The Frankl number of a lattice. St000699The toughness times the least common multiple of 1,. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000422The energy of a graph, if it is integral. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. 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. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001845The number of join irreducibles minus the rank of a lattice. St001626The number of maximal proper sublattices of a lattice.