Your data matches 370 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000010
(load all 2 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000010: 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)
 => [1,1]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([],6)
 => []
 => 0
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => 2
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 3
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => 3
([],7)
 => []
 => 0
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => 2
([(3,6),(4,6),(5,6)],7)
 => [1,1,1]
 => 3
([(3,6),(4,5)],7)
 => [1,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [1,1,1]
 => 3
([(2,3),(4,6),(5,6)],7)
 => [1,1,1]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => 3
Description
The length of the partition.
Matching statistic: St000148
(load all 3 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000148: 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)
 => [1,1]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([],6)
 => []
 => 0
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => 2
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 3
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => 3
([],7)
 => []
 => 0
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => 2
([(3,6),(4,6),(5,6)],7)
 => [1,1,1]
 => 3
([(3,6),(4,5)],7)
 => [1,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [1,1,1]
 => 3
([(2,3),(4,6),(5,6)],7)
 => [1,1,1]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => 3
Description
The number of odd parts of a partition.
Matching statistic: St000160
(load all 3 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000160: 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)
 => [1,1]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([],6)
 => []
 => 0
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => 2
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 3
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => 3
([],7)
 => []
 => 0
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => 2
([(3,6),(4,6),(5,6)],7)
 => [1,1,1]
 => 3
([(3,6),(4,5)],7)
 => [1,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [1,1,1]
 => 3
([(2,3),(4,6),(5,6)],7)
 => [1,1,1]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => 3
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000548
(load all 2 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000548: 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)
 => [1,1]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([],6)
 => []
 => 0
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => 2
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => 3
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => 3
([],7)
 => []
 => 0
([(5,6)],7)
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => 2
([(3,6),(4,6),(5,6)],7)
 => [1,1,1]
 => 3
([(3,6),(4,5)],7)
 => [1,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [1,1,1]
 => 3
([(2,3),(4,6),(5,6)],7)
 => [1,1,1]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => 3
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St001251
Mapping
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 0
([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => [2]
 => 1
([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => [2,1]
 => 1
([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => [1,1,1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
([],5)
 => [1,1,1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(1,4),(2,3)],5)
 => [2,2,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([],6)
 => [1,1,1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => 2
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => 3
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => 3
([],7)
 => [1,1,1,1,1,1,1]
 => 0
([(5,6)],7)
 => [2,1,1,1,1,1]
 => 1
([(4,6),(5,6)],7)
 => [2,2,1,1,1,1]
 => 2
([(3,6),(4,6),(5,6)],7)
 => [2,2,2,1,1,1]
 => 3
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [2,2,2,1,1,1]
 => 3
([(2,3),(4,6),(5,6)],7)
 => [2,2,2,1,1]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => 3
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001280
(load all 3 compositions to match this statistic)
Mapping
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => 0
([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => [2]
 => 1
([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => [2,1]
 => 1
([(0,2),(1,2)],3)
 => [2,2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => [1,1,1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
([],5)
 => [1,1,1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 3
([(1,4),(2,3)],5)
 => [2,2,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 1
([],6)
 => [1,1,1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => 2
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => 3
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => 3
([],7)
 => [1,1,1,1,1,1,1]
 => 0
([(5,6)],7)
 => [2,1,1,1,1,1]
 => 1
([(4,6),(5,6)],7)
 => [2,2,1,1,1,1]
 => 2
([(3,6),(4,6),(5,6)],7)
 => [2,2,2,1,1,1]
 => 3
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [2,2,2,1,1,1]
 => 3
([(2,3),(4,6),(5,6)],7)
 => [2,2,2,1,1]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => 3
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001613
(load all 2 compositions to match this statistic)
Mapping
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001613: 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)
 => 1
([],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),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([],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),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([],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),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 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),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(2,3),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
Description
The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001617
(load all 2 compositions to match this statistic)
Mapping
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001617: 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)
 => 1
([],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),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([],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),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([],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),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(0,5),(1,4),(2,3)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 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),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(3,6),(4,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(2,3),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 3
Description
The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying $$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: 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]
 => 1
([],3)
 => []
 => []
 => 0
([(1,2)],3)
 => [1]
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [1,1]
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([],4)
 => []
 => []
 => 0
([(2,3)],4)
 => [1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => [2]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [3]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [3]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([],5)
 => []
 => []
 => 0
([(3,4)],5)
 => [1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => [2]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [3]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [3]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([],6)
 => []
 => []
 => 0
([(4,5)],6)
 => [1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => [2]
 => 2
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => 3
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [3]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [3]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [3]
 => 3
([],7)
 => []
 => []
 => 0
([(5,6)],7)
 => [1]
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => [2]
 => 2
([(3,6),(4,6),(5,6)],7)
 => [1,1,1]
 => [3]
 => 3
([(3,6),(4,5)],7)
 => [1,1]
 => [2]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [1,1,1]
 => [3]
 => 3
([(2,3),(4,6),(5,6)],7)
 => [1,1,1]
 => [3]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3]
 => [1,1,1]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [3]
 => 3
Description
The largest part of an integer partition.
Matching statistic: St000378
(load all 2 compositions to match this statistic)
Mapping
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: 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]
 => 1
([],3)
 => []
 => []
 => 0
([(1,2)],3)
 => [1]
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [1,1]
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([],4)
 => []
 => []
 => 0
([(2,3)],4)
 => [1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [1,1]
 => [2]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [2,1]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [2,1]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([],5)
 => []
 => []
 => 0
([(3,4)],5)
 => [1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [1,1]
 => [2]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [2,1]
 => 3
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [2,1]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [2,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([],6)
 => []
 => []
 => 0
([(4,5)],6)
 => [1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [1,1]
 => [2]
 => 2
([(2,5),(3,5),(4,5)],6)
 => [1,1,1]
 => [2,1]
 => 3
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [1,1,1]
 => [2,1]
 => 3
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [2,1]
 => 3
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [2,1]
 => 3
([],7)
 => []
 => []
 => 0
([(5,6)],7)
 => [1]
 => [1]
 => 1
([(4,6),(5,6)],7)
 => [1,1]
 => [2]
 => 2
([(3,6),(4,6),(5,6)],7)
 => [1,1,1]
 => [2,1]
 => 3
([(3,6),(4,5)],7)
 => [1,1]
 => [2]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [1,1,1]
 => [2,1]
 => 3
([(2,3),(4,6),(5,6)],7)
 => [1,1,1]
 => [2,1]
 => 3
([(4,5),(4,6),(5,6)],7)
 => [3]
 => [1,1,1]
 => 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [2,1]
 => 3
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
The following 360 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001615The number of join prime elements of a lattice. St001622The number of join-irreducible elements of a lattice. St001814The number of partitions interlacing the given partition. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001933The largest multiplicity of a part in an integer partition. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000288The number of ones in a binary word. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000475The number of parts equal to 1 in a partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. 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. 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. St000982The length of the longest constant subword. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in 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. St001249Sum of the odd parts of a partition. St001267The length of the Lyndon factorization of the binary word. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. 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. St001437The flex of a binary word. St001485The modular major index of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001884The number of borders of a binary word. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000012The area of a Dyck path. St000228The size of a partition. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. 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. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001127The sum of the squares of the parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000005The bounce statistic of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000120The number of left tunnels of a Dyck path. St000293The number of inversions of a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000877The depth of the binary word interpreted as a path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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: 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001462The number of factors of a standard tableaux under concatenation. St001480The number of simple summands of the module J^2/J^3. St001481The minimal height of a peak of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer partition. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. 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. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000006The dinv of a Dyck path. St000053The number of valleys of the Dyck path. St000117The number of centered tunnels of a Dyck path. St000306The bounce count of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. 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. St000674The number of hills of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000921The number of internal inversions of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. 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 $c_{n-1}$. 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. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. 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. St001172The number of 1-rises at odd height of a Dyck path. 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 $eA$. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001523The degree of symmetry of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001910The height of the middle non-run of a Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000993The multiplicity of the largest part of an integer partition. 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. St000477The weight of a partition according to Alladi. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001621The number of atoms of a lattice. 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. St000420The number of Dyck paths that are weakly above a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001808The box weight or horizontal decoration of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000928The sum of the coefficients of the character polynomial of an integer partition. St000454The largest eigenvalue of a graph if it is integral. 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. 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. 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. 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 $eA$. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000145The Dyson rank of a partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000640The rank of the largest boolean interval in a poset. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. St000706The product of the factorials of the multiplicities of an integer partition. 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. St001568The smallest positive integer that does not appear twice in the partition. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. 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. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000509The diagonal index (content) of a partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001060The distinguishing index of a graph. 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. St001645The pebbling number of a connected graph. St000741The Colin de Verdière graph invariant. St001330The hat guessing number of a graph. 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. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001651The Frankl number of a lattice. St000260The radius of a connected graph. St000259The diameter 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000422The energy of a graph, if it is integral. St000264The girth of a graph, which is not a tree. St001118The acyclic chromatic index of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. 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. 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. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St000474Dyson's crank 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. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001943The sum of the squares of the hook lengths of an integer partition. St000379The number of Hamiltonian cycles in a graph. St001845The number of join irreducibles minus the rank of a lattice. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.