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Your data matches 719 different statistics following compositions of up to 3 maps.
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Matching statistic: St001535
(load all 116 compositions to match this statistic)
(load all 116 compositions to match this statistic)
St001535: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = -1 + 1
[1,2] => 0 = -1 + 1
[2,1] => 0 = -1 + 1
[1,2,3] => 0 = -1 + 1
[1,3,2] => 0 = -1 + 1
[2,1,3] => 0 = -1 + 1
[2,3,1] => 0 = -1 + 1
[3,1,2] => 0 = -1 + 1
[1,2,3,4] => 0 = -1 + 1
[1,2,4,3] => 0 = -1 + 1
[1,3,2,4] => 0 = -1 + 1
[1,3,4,2] => 0 = -1 + 1
[1,4,2,3] => 0 = -1 + 1
[2,1,3,4] => 0 = -1 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 0 = -1 + 1
[3,1,2,4] => 0 = -1 + 1
[1,2,3,4,5] => 0 = -1 + 1
[1,2,3,5,4] => 0 = -1 + 1
[1,2,4,3,5] => 0 = -1 + 1
[1,2,4,5,3] => 0 = -1 + 1
[1,2,5,3,4] => 0 = -1 + 1
[1,3,2,4,5] => 0 = -1 + 1
[1,3,2,5,4] => 2 = 1 + 1
[1,3,4,2,5] => 0 = -1 + 1
[1,4,2,3,5] => 0 = -1 + 1
[2,1,3,4,5] => 0 = -1 + 1
[2,1,3,5,4] => 2 = 1 + 1
[2,1,4,3,5] => 2 = 1 + 1
[2,3,1,4,5] => 0 = -1 + 1
[3,1,2,4,5] => 0 = -1 + 1
[1,2,3,4,5,6] => 0 = -1 + 1
[1,2,3,4,6,5] => 0 = -1 + 1
[1,2,3,5,4,6] => 0 = -1 + 1
[1,2,3,5,6,4] => 0 = -1 + 1
[1,2,4,3,5,6] => 0 = -1 + 1
[1,2,4,3,6,5] => 2 = 1 + 1
[1,2,4,5,3,6] => 0 = -1 + 1
[1,3,2,4,5,6] => 0 = -1 + 1
[1,3,2,4,6,5] => 2 = 1 + 1
[1,3,2,5,4,6] => 2 = 1 + 1
[1,3,4,2,5,6] => 0 = -1 + 1
[2,1,3,4,5,6] => 0 = -1 + 1
[2,1,3,4,6,5] => 2 = 1 + 1
[2,1,3,5,4,6] => 2 = 1 + 1
[2,1,4,3,5,6] => 2 = 1 + 1
[2,3,1,4,5,6] => 0 = -1 + 1
Description
The number of cyclic alignments of a permutation.
The pair $(i,j)$ is a cyclic alignment of a permutation $\pi$ if $i, j, \pi(j), \pi(i)$ are cyclically ordered and all distinct, see Section 5 of [1]
Matching statistic: St001536
(load all 155 compositions to match this statistic)
(load all 155 compositions to match this statistic)
St001536: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = -1 + 1
[1,2] => 0 = -1 + 1
[2,1] => 0 = -1 + 1
[1,2,3] => 0 = -1 + 1
[1,3,2] => 0 = -1 + 1
[2,1,3] => 0 = -1 + 1
[2,3,1] => 0 = -1 + 1
[3,1,2] => 0 = -1 + 1
[1,2,3,4] => 0 = -1 + 1
[1,2,4,3] => 0 = -1 + 1
[1,3,2,4] => 0 = -1 + 1
[1,3,4,2] => 0 = -1 + 1
[1,4,2,3] => 0 = -1 + 1
[2,1,3,4] => 0 = -1 + 1
[2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => 0 = -1 + 1
[3,1,2,4] => 0 = -1 + 1
[1,2,3,4,5] => 0 = -1 + 1
[1,2,3,5,4] => 0 = -1 + 1
[1,2,4,3,5] => 0 = -1 + 1
[1,2,4,5,3] => 0 = -1 + 1
[1,2,5,3,4] => 0 = -1 + 1
[1,3,2,4,5] => 0 = -1 + 1
[1,3,2,5,4] => 2 = 1 + 1
[1,3,4,2,5] => 0 = -1 + 1
[1,4,2,3,5] => 0 = -1 + 1
[2,1,3,4,5] => 0 = -1 + 1
[2,1,3,5,4] => 2 = 1 + 1
[2,1,4,3,5] => 2 = 1 + 1
[2,3,1,4,5] => 0 = -1 + 1
[3,1,2,4,5] => 0 = -1 + 1
[1,2,3,4,5,6] => 0 = -1 + 1
[1,2,3,4,6,5] => 0 = -1 + 1
[1,2,3,5,4,6] => 0 = -1 + 1
[1,2,3,5,6,4] => 0 = -1 + 1
[1,2,4,3,5,6] => 0 = -1 + 1
[1,2,4,3,6,5] => 2 = 1 + 1
[1,2,4,5,3,6] => 0 = -1 + 1
[1,3,2,4,5,6] => 0 = -1 + 1
[1,3,2,4,6,5] => 2 = 1 + 1
[1,3,2,5,4,6] => 2 = 1 + 1
[1,3,4,2,5,6] => 0 = -1 + 1
[2,1,3,4,5,6] => 0 = -1 + 1
[2,1,3,4,6,5] => 2 = 1 + 1
[2,1,3,5,4,6] => 2 = 1 + 1
[2,1,4,3,5,6] => 2 = 1 + 1
[2,3,1,4,5,6] => 0 = -1 + 1
Description
The number of cyclic misalignments of a permutation.
The pair $(i,j)$ is a cyclic alignment of a permutation $\pi$ if $i, \pi(i), j, \pi(j)$ are cyclically ordered and all distinct, see Section 5 of [1].
Matching statistic: St000039
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = -1 + 1
[1,2] => [1,2] => [1,2] => 0 = -1 + 1
[2,1] => [2,1] => [2,1] => 0 = -1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = -1 + 1
[1,3,2] => [3,1,2] => [3,2,1] => 0 = -1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => 0 = -1 + 1
[2,3,1] => [2,3,1] => [3,2,1] => 0 = -1 + 1
[3,1,2] => [1,3,2] => [1,3,2] => 0 = -1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = -1 + 1
[1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0 = -1 + 1
[1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0 = -1 + 1
[1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 0 = -1 + 1
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0 = -1 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = -1 + 1
[2,1,4,3] => [2,4,1,3] => [3,4,1,2] => 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 0 = -1 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = -1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = -1 + 1
[1,2,3,5,4] => [5,1,2,3,4] => [5,2,3,4,1] => 0 = -1 + 1
[1,2,4,3,5] => [4,1,2,3,5] => [4,2,3,1,5] => 0 = -1 + 1
[1,2,4,5,3] => [4,5,1,2,3] => [5,4,3,2,1] => 0 = -1 + 1
[1,2,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => 0 = -1 + 1
[1,3,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 0 = -1 + 1
[1,3,2,5,4] => [3,5,1,2,4] => [4,5,3,1,2] => 2 = 1 + 1
[1,3,4,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 0 = -1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0 = -1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = -1 + 1
[2,1,3,5,4] => [2,5,1,3,4] => [3,5,1,4,2] => 2 = 1 + 1
[2,1,4,3,5] => [2,4,1,3,5] => [3,4,1,2,5] => 2 = 1 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 0 = -1 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0 = -1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = -1 + 1
[1,2,3,4,6,5] => [6,1,2,3,4,5] => [6,2,3,4,5,1] => 0 = -1 + 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,2,3,4,1,6] => 0 = -1 + 1
[1,2,3,5,6,4] => [5,6,1,2,3,4] => [6,5,3,4,2,1] => 0 = -1 + 1
[1,2,4,3,5,6] => [4,1,2,3,5,6] => [4,2,3,1,5,6] => 0 = -1 + 1
[1,2,4,3,6,5] => [4,6,1,2,3,5] => [5,6,3,4,1,2] => 2 = 1 + 1
[1,2,4,5,3,6] => [4,5,1,2,3,6] => [5,4,3,2,1,6] => 0 = -1 + 1
[1,3,2,4,5,6] => [3,1,2,4,5,6] => [3,2,1,4,5,6] => 0 = -1 + 1
[1,3,2,4,6,5] => [3,6,1,2,4,5] => [4,6,3,1,5,2] => 2 = 1 + 1
[1,3,2,5,4,6] => [3,5,1,2,4,6] => [4,5,3,1,2,6] => 2 = 1 + 1
[1,3,4,2,5,6] => [3,4,1,2,5,6] => [4,3,2,1,5,6] => 0 = -1 + 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0 = -1 + 1
[2,1,3,4,6,5] => [2,6,1,3,4,5] => [3,6,1,4,5,2] => 2 = 1 + 1
[2,1,3,5,4,6] => [2,5,1,3,4,6] => [3,5,1,4,2,6] => 2 = 1 + 1
[2,1,4,3,5,6] => [2,4,1,3,5,6] => [3,4,1,2,5,6] => 2 = 1 + 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [3,2,1,4,5,6] => 0 = -1 + 1
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000143
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0 = -1 + 1
[1,2] => ([(0,1)],2)
=> [1]
=> 0 = -1 + 1
[2,1] => ([],2)
=> [2]
=> 0 = -1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> [1]
=> 0 = -1 + 1
[1,3,2] => ([(0,1),(0,2)],3)
=> [2]
=> 0 = -1 + 1
[2,1,3] => ([(0,2),(1,2)],3)
=> [2]
=> 0 = -1 + 1
[2,3,1] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[3,1,2] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = -1 + 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 0 = -1 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = -1 + 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 0 = -1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0 = -1 + 1
[1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> 0 = -1 + 1
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = -1 + 1
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St000185
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0 = -1 + 1
[1,2] => ([(0,1)],2)
=> [1]
=> 0 = -1 + 1
[2,1] => ([],2)
=> [2]
=> 0 = -1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> [1]
=> 0 = -1 + 1
[1,3,2] => ([(0,1),(0,2)],3)
=> [2]
=> 0 = -1 + 1
[2,1,3] => ([(0,2),(1,2)],3)
=> [2]
=> 0 = -1 + 1
[2,3,1] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[3,1,2] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = -1 + 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 0 = -1 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = -1 + 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 0 = -1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0 = -1 + 1
[1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> 0 = -1 + 1
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = -1 + 1
Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$
\sum_i \binom{\lambda^{\prime}_i}{2}
$$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Matching statistic: St000749
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0 = -1 + 1
[1,2] => ([(0,1)],2)
=> [1]
=> 0 = -1 + 1
[2,1] => ([],2)
=> [2]
=> 0 = -1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> [1]
=> 0 = -1 + 1
[1,3,2] => ([(0,1),(0,2)],3)
=> [2]
=> 0 = -1 + 1
[2,1,3] => ([(0,2),(1,2)],3)
=> [2]
=> 0 = -1 + 1
[2,3,1] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[3,1,2] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = -1 + 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 0 = -1 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = -1 + 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 0 = -1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0 = -1 + 1
[1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> 0 = -1 + 1
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = -1 + 1
Description
The 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.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St000752
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 0 = -1 + 1
[1,2] => [1,1]
=> [2]
=> 0 = -1 + 1
[2,1] => [2]
=> [1,1]
=> 0 = -1 + 1
[1,2,3] => [1,1,1]
=> [2,1]
=> 0 = -1 + 1
[1,3,2] => [2,1]
=> [1,1,1]
=> 0 = -1 + 1
[2,1,3] => [2,1]
=> [1,1,1]
=> 0 = -1 + 1
[2,3,1] => [2,1]
=> [1,1,1]
=> 0 = -1 + 1
[3,1,2] => [2,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,2,3,4] => [1,1,1,1]
=> [2,2]
=> 0 = -1 + 1
[1,2,4,3] => [2,1,1]
=> [2,1,1]
=> 0 = -1 + 1
[1,3,2,4] => [2,1,1]
=> [2,1,1]
=> 0 = -1 + 1
[1,3,4,2] => [2,1,1]
=> [2,1,1]
=> 0 = -1 + 1
[1,4,2,3] => [2,1,1]
=> [2,1,1]
=> 0 = -1 + 1
[2,1,3,4] => [2,1,1]
=> [2,1,1]
=> 0 = -1 + 1
[2,1,4,3] => [2,2]
=> [4]
=> 2 = 1 + 1
[2,3,1,4] => [2,1,1]
=> [2,1,1]
=> 0 = -1 + 1
[3,1,2,4] => [2,1,1]
=> [2,1,1]
=> 0 = -1 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [2,2,1]
=> 0 = -1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[1,2,4,5,3] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[1,2,5,3,4] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[1,3,2,5,4] => [2,2,1]
=> [4,1]
=> 2 = 1 + 1
[1,3,4,2,5] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[1,4,2,3,5] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[2,1,3,5,4] => [2,2,1]
=> [4,1]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [4,1]
=> 2 = 1 + 1
[2,3,1,4,5] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[3,1,2,4,5] => [2,1,1,1]
=> [2,1,1,1]
=> 0 = -1 + 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [2,2,2]
=> 0 = -1 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
[1,2,3,5,6,4] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
[1,2,4,3,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,2,4,5,3,6] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
[1,3,2,4,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,3,2,5,4,6] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,3,4,2,5,6] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
[2,1,3,4,5,6] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
[2,1,3,4,6,5] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[2,1,3,5,4,6] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[2,1,4,3,5,6] => [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[2,3,1,4,5,6] => [2,1,1,1,1]
=> [2,2,1,1]
=> 0 = -1 + 1
Description
The Grundy value for the game 'Couples are forever' on an integer partition.
Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
Matching statistic: St000995
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000995: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000995: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> 0 = -1 + 1
[1,2] => [1,1]
=> [1]
=> 0 = -1 + 1
[2,1] => [2]
=> []
=> 0 = -1 + 1
[1,2,3] => [1,1,1]
=> [1,1]
=> 0 = -1 + 1
[1,3,2] => [2,1]
=> [1]
=> 0 = -1 + 1
[2,1,3] => [2,1]
=> [1]
=> 0 = -1 + 1
[2,3,1] => [2,1]
=> [1]
=> 0 = -1 + 1
[3,1,2] => [2,1]
=> [1]
=> 0 = -1 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 0 = -1 + 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 0 = -1 + 1
[1,3,4,2] => [2,1,1]
=> [1,1]
=> 0 = -1 + 1
[1,4,2,3] => [2,1,1]
=> [1,1]
=> 0 = -1 + 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 0 = -1 + 1
[2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[2,3,1,4] => [2,1,1]
=> [1,1]
=> 0 = -1 + 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> 0 = -1 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[3,1,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0 = -1 + 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0 = -1 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[1,2,3,5,6,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,2,4,5,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,3,4,2,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[2,1,3,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
[2,1,3,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[2,1,3,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[2,1,4,3,5,6] => [2,2,1,1]
=> [2,1,1]
=> 2 = 1 + 1
[2,3,1,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = -1 + 1
Description
The largest even part of an integer partition.
Matching statistic: St001176
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0 = -1 + 1
[1,2] => ([(0,1)],2)
=> [1]
=> 0 = -1 + 1
[2,1] => ([],2)
=> [2]
=> 0 = -1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> [1]
=> 0 = -1 + 1
[1,3,2] => ([(0,1),(0,2)],3)
=> [2]
=> 0 = -1 + 1
[2,1,3] => ([(0,2),(1,2)],3)
=> [2]
=> 0 = -1 + 1
[2,3,1] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[3,1,2] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = -1 + 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 0 = -1 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = -1 + 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 0 = -1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0 = -1 + 1
[1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> 0 = -1 + 1
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = -1 + 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001214
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0 = -1 + 1
[1,2] => ([(0,1)],2)
=> [1]
=> 0 = -1 + 1
[2,1] => ([],2)
=> [2]
=> 0 = -1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> [1]
=> 0 = -1 + 1
[1,3,2] => ([(0,1),(0,2)],3)
=> [2]
=> 0 = -1 + 1
[2,1,3] => ([(0,2),(1,2)],3)
=> [2]
=> 0 = -1 + 1
[2,3,1] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[3,1,2] => ([(1,2)],3)
=> [3]
=> 0 = -1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = -1 + 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 0 = -1 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = -1 + 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0 = -1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = -1 + 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 0 = -1 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 0 = -1 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0 = -1 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = -1 + 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0 = -1 + 1
[1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 0 = -1 + 1
[1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> 0 = -1 + 1
[1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 0 = -1 + 1
[1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> 0 = -1 + 1
[1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> 0 = -1 + 1
[2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0 = -1 + 1
[2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = -1 + 1
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
The following 709 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001961The sum of the greatest common divisors of all pairs of parts. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St001564The value of the forgotten symmetric functions when all variables set to 1. St000511The number of invariant subsets when acting with a permutation of given cycle type. St001472The permanent of the Coxeter matrix of the poset. St000008The major index of the composition. St000027The major index of a Dyck path. St000142The number of even parts of a partition. St000154The sum of the descent bottoms of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000223The number of nestings in the permutation. St000272The treewidth of a graph. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000362The size of a minimal vertex cover of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000536The pathwidth of a graph. St000552The number of cut vertices of a graph. St000648The number of 2-excedences of a permutation. St000979Half of MacMahon's equal index of a Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001308The number of induced paths on three vertices in a graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001374The Padmakar-Ivan index of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001623The number of doubly irreducible elements of a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001657The number of twos in an integer partition. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001695The natural comajor index of a standard Young tableau. St001696The natural major index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001792The arboricity of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001812The biclique partition number of a graph. St001902The number of potential covers of a poset. St000001The number of reduced words for a permutation. St000047The number of standard immaculate tableaux of a given shape. St000096The number of spanning trees of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000182The number of permutations whose cycle type is the given integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000267The number of maximal spanning forests contained in a graph. St000271The chromatic index of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000763The sum of the positions of the strong records of an integer composition. 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. St000822The Hadwiger number of the graph. St000847The number of standard Young tableaux whose descent set is the binary word. St000935The number of ordered refinements of an integer partition. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001546The number of monomials in the Tutte polynomial of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001694The number of maximal dissociation sets in a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001850The number of Hecke atoms of a permutation. St001963The tree-depth of a graph. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000290The major index of a binary word. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000538The number of even inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001485The modular major index of a binary word. St001644The dimension of a graph. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000762The sum of the positions of the weak records of an integer composition. St000988The orbit size of a permutation under Foata's bijection. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St000219The number of occurrences of the pattern 231 in a permutation. St001330The hat guessing number of a graph. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St000910The number of maximal chains of minimal length in a poset. St000656The number of cuts of a poset. St000146The Andrews-Garvan crank of a partition. St000311The number of vertices of odd degree in a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000350The sum of the vertex degrees of a graph. St000422The energy of a graph, if it is integral. St000465The first Zagreb index of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000571The F-index (or forgotten topological index) of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000915The Ore degree of a graph. St000934The 2-degree of an integer partition. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001956The comajor index for set-valued two-row standard Young tableaux. St000347The inversion sum of a binary word. St000456The monochromatic index of a connected graph. St000526The number of posets with combinatorially isomorphic order polytopes. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001118The acyclic chromatic index of a graph. St001182Number of indecomposable injective modules with codominant dimension at least two 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. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001303The number of dominating sets of vertices of a graph. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001721The degree of a binary word. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001930The weak major index of a binary word. St000289The decimal representation of a binary word. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. 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. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001669The number of single rises in a Dyck path. St001885The number of binary words with the same proper border set. St000438The position of the last up step in a Dyck path. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001586The number of odd parts smaller than the largest even part in an integer partition. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001856The number of edges in the reduced word graph of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000068The number of minimal elements in a poset. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. 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. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000509The diagonal index (content) of a partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000467The hyper-Wiener index of a connected graph. St001964The interval resolution global dimension of a poset. 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 $eA$. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001720The minimal length of a chain of small intervals in a lattice. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001890The maximum magnitude of the Möbius function of a poset. St000741The Colin de Verdière graph invariant. St000914The sum of the values of the Möbius function of a poset. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001768The number of reduced words of a signed permutation. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000264The girth of a graph, which is not a tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000095The number of triangles of a graph. St000137The Grundy value of an integer partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St000449The number of pairs of vertices of a graph with distance 4. St001396Number of triples of incomparable elements in a finite poset. St001525The number of symmetric hooks on the diagonal of a partition. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001871The number of triconnected components of a graph. St001939The number of parts that are equal to their multiplicity in the integer partition. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001518The number of graphs with the same ordinary spectrum as the given graph. St001527The cyclic permutation representation number of an integer partition. St000225Difference between largest and smallest parts in a partition. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001645The pebbling number of a connected graph. St000981The length of the longest zigzag subpath. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001621The number of atoms of a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000477The weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000928The sum of the coefficients of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000997The even-odd crank 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. 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. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000444The length of the maximal rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001531Number of partial orders contained in the poset determined by the Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001959The product of the heights of the peaks of a Dyck path. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. 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. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001480The number of simple summands of the module J^2/J^3. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001808The box weight or horizontal decoration of a Dyck path. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001875The number of simple modules with projective dimension at most 1. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000907The number of maximal antichains of minimal length in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001857The number of edges in the reduced word graph of a signed permutation. St000657The smallest part of an integer composition. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000077The number of boxed and circled entries. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000268The number of strongly connected orientations of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000295The length of the border of a binary word. St000344The number of strongly connected outdegree sequences of a graph. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000637The length of the longest cycle in a graph. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000948The chromatic discriminant of a graph. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. 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. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001367The smallest number which does not occur as degree of a vertex in a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001513The number of nested exceedences of a permutation. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001736The total number of cycles in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000069The number of maximal elements of a poset. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000535The rank-width of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000544The cop number of a graph. St000553The number of blocks of a graph. St000570The Edelman-Greene number of a permutation. St000640The rank of the largest boolean interval in a poset. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001282The number of graphs with the same chromatic polynomial. St001333The cardinality of a minimal edge-isolating set of a graph. St001344The neighbouring number of a permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001363The Euler characteristic of a graph according to Knill. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001743The discrepancy of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000061The number of nodes on the left branch of a binary tree. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000364The exponent of the automorphism group of a graph. St000401The size of the symmetry class of a permutation. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St000917The open packing number of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001399The distinguishing number of a poset. St001654The monophonic hull number of a graph. St001716The 1-improper chromatic number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000102The charge of a semistandard tableau. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000315The number of isolated vertices of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St000873The aix statistic of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000990The first ascent of a permutation. St000474Dyson's crank of a partition. St000145The Dyson rank of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000466The Gutman (or modified Schultz) index of a connected graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000782The indicator function of whether a given perfect matching is an L & P matching. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. 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. 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. St001571The Cartan determinant of the integer partition. 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. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. 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. St001609The number of coloured 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. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. 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. 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. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000806The semiperimeter of the associated bargraph. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001409The maximal entry of a semistandard tableau.
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