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Your data matches 366 different statistics following compositions of up to 3 maps.
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Matching statistic: St001298
(load all 109 compositions to match this statistic)
(load all 109 compositions to match this statistic)
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 2 - 2
[1,2] => 1 = 3 - 2
[2,1] => 0 = 2 - 2
[1,2,3] => 2 = 4 - 2
[1,3,2] => 1 = 3 - 2
[2,1,3] => 1 = 3 - 2
[2,3,1] => 1 = 3 - 2
[3,1,2] => 1 = 3 - 2
[1,2,3,4] => 3 = 5 - 2
[1,2,4,3] => 2 = 4 - 2
[1,3,2,4] => 2 = 4 - 2
[1,3,4,2] => 2 = 4 - 2
[1,4,2,3] => 2 = 4 - 2
[2,1,3,4] => 2 = 4 - 2
[2,1,4,3] => 2 = 4 - 2
[2,3,1,4] => 2 = 4 - 2
[3,1,2,4] => 2 = 4 - 2
[1,2,3,4,5] => 4 = 6 - 2
[1,2,3,5,4] => 3 = 5 - 2
[1,2,4,3,5] => 3 = 5 - 2
[1,3,2,4,5] => 3 = 5 - 2
[2,1,3,4,5] => 3 = 5 - 2
[1,2,3,4,5,6] => 5 = 7 - 2
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Matching statistic: St001389
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1 = 2 - 1
[1,2] => [2]
=> 2 = 3 - 1
[2,1] => [1,1]
=> 1 = 2 - 1
[1,2,3] => [3]
=> 3 = 4 - 1
[1,3,2] => [2,1]
=> 2 = 3 - 1
[2,1,3] => [2,1]
=> 2 = 3 - 1
[2,3,1] => [2,1]
=> 2 = 3 - 1
[3,1,2] => [2,1]
=> 2 = 3 - 1
[1,2,3,4] => [4]
=> 4 = 5 - 1
[1,2,4,3] => [3,1]
=> 3 = 4 - 1
[1,3,2,4] => [3,1]
=> 3 = 4 - 1
[1,3,4,2] => [3,1]
=> 3 = 4 - 1
[1,4,2,3] => [3,1]
=> 3 = 4 - 1
[2,1,3,4] => [3,1]
=> 3 = 4 - 1
[2,1,4,3] => [2,2]
=> 3 = 4 - 1
[2,3,1,4] => [3,1]
=> 3 = 4 - 1
[3,1,2,4] => [3,1]
=> 3 = 4 - 1
[1,2,3,4,5] => [5]
=> 5 = 6 - 1
[1,2,3,5,4] => [4,1]
=> 4 = 5 - 1
[1,2,4,3,5] => [4,1]
=> 4 = 5 - 1
[1,3,2,4,5] => [4,1]
=> 4 = 5 - 1
[2,1,3,4,5] => [4,1]
=> 4 = 5 - 1
[1,2,3,4,5,6] => [6]
=> 6 = 7 - 1
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000245
(load all 139 compositions to match this statistic)
(load all 139 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 1 = 3 - 2
[2,1] => [2,1] => 0 = 2 - 2
[1,2,3] => [1,2,3] => 2 = 4 - 2
[1,3,2] => [3,1,2] => 1 = 3 - 2
[2,1,3] => [2,1,3] => 1 = 3 - 2
[2,3,1] => [1,3,2] => 1 = 3 - 2
[3,1,2] => [2,3,1] => 1 = 3 - 2
[1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,2,4,3] => [4,1,2,3] => 2 = 4 - 2
[1,3,2,4] => [3,1,2,4] => 2 = 4 - 2
[1,3,4,2] => [2,4,1,3] => 2 = 4 - 2
[1,4,2,3] => [3,4,1,2] => 2 = 4 - 2
[2,1,3,4] => [2,1,3,4] => 2 = 4 - 2
[2,1,4,3] => [1,4,2,3] => 2 = 4 - 2
[2,3,1,4] => [1,3,2,4] => 2 = 4 - 2
[3,1,2,4] => [2,3,1,4] => 2 = 4 - 2
[1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,2,3,5,4] => [5,1,2,3,4] => 3 = 5 - 2
[1,2,4,3,5] => [4,1,2,3,5] => 3 = 5 - 2
[1,3,2,4,5] => [3,1,2,4,5] => 3 = 5 - 2
[2,1,3,4,5] => [2,1,3,4,5] => 3 = 5 - 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 5 = 7 - 2
Description
The number of ascents of a permutation.
Matching statistic: St000662
(load all 94 compositions to match this statistic)
(load all 94 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [2,1] => 1 = 3 - 2
[2,1] => [1,2] => 0 = 2 - 2
[1,2,3] => [3,2,1] => 2 = 4 - 2
[1,3,2] => [3,1,2] => 1 = 3 - 2
[2,1,3] => [2,3,1] => 1 = 3 - 2
[2,3,1] => [2,1,3] => 1 = 3 - 2
[3,1,2] => [1,3,2] => 1 = 3 - 2
[1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,2,4,3] => [4,3,1,2] => 2 = 4 - 2
[1,3,2,4] => [4,2,3,1] => 2 = 4 - 2
[1,3,4,2] => [4,2,1,3] => 2 = 4 - 2
[1,4,2,3] => [4,1,3,2] => 2 = 4 - 2
[2,1,3,4] => [3,4,2,1] => 2 = 4 - 2
[2,1,4,3] => [3,4,1,2] => 2 = 4 - 2
[2,3,1,4] => [3,2,4,1] => 2 = 4 - 2
[3,1,2,4] => [2,4,3,1] => 2 = 4 - 2
[1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,2,3,5,4] => [5,4,3,1,2] => 3 = 5 - 2
[1,2,4,3,5] => [5,4,2,3,1] => 3 = 5 - 2
[1,3,2,4,5] => [5,3,4,2,1] => 3 = 5 - 2
[2,1,3,4,5] => [4,5,3,2,1] => 3 = 5 - 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => 5 = 7 - 2
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000672
(load all 218 compositions to match this statistic)
(load all 218 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 1 = 3 - 2
[2,1] => [2,1] => 0 = 2 - 2
[1,2,3] => [1,2,3] => 2 = 4 - 2
[1,3,2] => [3,1,2] => 1 = 3 - 2
[2,1,3] => [2,1,3] => 1 = 3 - 2
[2,3,1] => [1,3,2] => 1 = 3 - 2
[3,1,2] => [2,3,1] => 1 = 3 - 2
[1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,2,4,3] => [4,1,2,3] => 2 = 4 - 2
[1,3,2,4] => [3,1,2,4] => 2 = 4 - 2
[1,3,4,2] => [2,4,1,3] => 2 = 4 - 2
[1,4,2,3] => [3,4,1,2] => 2 = 4 - 2
[2,1,3,4] => [2,1,3,4] => 2 = 4 - 2
[2,1,4,3] => [1,4,2,3] => 2 = 4 - 2
[2,3,1,4] => [1,3,2,4] => 2 = 4 - 2
[3,1,2,4] => [2,3,1,4] => 2 = 4 - 2
[1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,2,3,5,4] => [5,1,2,3,4] => 3 = 5 - 2
[1,2,4,3,5] => [4,1,2,3,5] => 3 = 5 - 2
[1,3,2,4,5] => [3,1,2,4,5] => 3 = 5 - 2
[2,1,3,4,5] => [2,1,3,4,5] => 3 = 5 - 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 5 = 7 - 2
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001176
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0 = 2 - 2
[1,2] => [1,1]
=> 1 = 3 - 2
[2,1] => [2]
=> 0 = 2 - 2
[1,2,3] => [1,1,1]
=> 2 = 4 - 2
[1,3,2] => [2,1]
=> 1 = 3 - 2
[2,1,3] => [2,1]
=> 1 = 3 - 2
[2,3,1] => [2,1]
=> 1 = 3 - 2
[3,1,2] => [2,1]
=> 1 = 3 - 2
[1,2,3,4] => [1,1,1,1]
=> 3 = 5 - 2
[1,2,4,3] => [2,1,1]
=> 2 = 4 - 2
[1,3,2,4] => [2,1,1]
=> 2 = 4 - 2
[1,3,4,2] => [2,1,1]
=> 2 = 4 - 2
[1,4,2,3] => [2,1,1]
=> 2 = 4 - 2
[2,1,3,4] => [2,1,1]
=> 2 = 4 - 2
[2,1,4,3] => [2,2]
=> 2 = 4 - 2
[2,3,1,4] => [2,1,1]
=> 2 = 4 - 2
[3,1,2,4] => [2,1,1]
=> 2 = 4 - 2
[1,2,3,4,5] => [1,1,1,1,1]
=> 4 = 6 - 2
[1,2,3,5,4] => [2,1,1,1]
=> 3 = 5 - 2
[1,2,4,3,5] => [2,1,1,1]
=> 3 = 5 - 2
[1,3,2,4,5] => [2,1,1,1]
=> 3 = 5 - 2
[2,1,3,4,5] => [2,1,1,1]
=> 3 = 5 - 2
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 5 = 7 - 2
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: St001211
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 2
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 3
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001321
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([(0,1)],2)
=> 2
[1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 3
[2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 4
[1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[1,2,4,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,2,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7
Description
The number of vertices of the largest induced subforest of a graph.
Matching statistic: St001492
(load all 59 compositions to match this statistic)
(load all 59 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 2
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 3
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 4
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 5
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 4
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St001626
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(load all 11 compositions to match this statistic)
Values
[1] => ([],1)
=> ([(0,1)],2)
=> 2
[1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
[2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of maximal proper sublattices of a lattice.
The following 356 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000062The length of the longest increasing subsequence of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000532The total number of rook placements on a Ferrers board. St000738The first entry in the last row of a standard tableau. St000822The Hadwiger number of the graph. St001116The game chromatic number of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001286The annihilation number of a graph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000211The rank of the set partition. St000228The size of a partition. St000272The treewidth of a graph. St000331The number of upper interactions of a Dyck path. St000362The size of a minimal vertex cover of a graph. 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. St000536The pathwidth of a graph. St000784The maximum of the length and the largest part of the integer partition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001644The dimension of a graph. St001962The proper pathwidth of a graph. 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 $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001746The coalition number of a graph. St000007The number of saliances of the permutation. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000054The first entry of the permutation. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000147The largest part of an integer partition. St000167The number of leaves of an ordered tree. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000288The number of ones in a binary word. St000293The number of inversions of a binary word. St000314The number of left-to-right-maxima of a permutation. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000383The last part of an integer composition. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000482The (zero)-forcing number of a graph. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000912The number of maximal antichains in a poset. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama 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:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001497The position of the largest weak excedence of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000012The area of a Dyck path. St000013The height of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000053The number of valleys of the Dyck path. St000067The inversion number of the alternating sign matrix. St000074The number of special entries. St000080The rank of the poset. St000141The maximum drop size of a permutation. St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000339The maf index of a permutation. St000369The dinv deficit of a Dyck path. St000441The number of successions of a permutation. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000703The number of deficiencies of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000921The number of internal inversions of a binary word. St000996The number of exclusive left-to-right maxima of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001127The sum of the squares of the parts of a partition. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001427The number of descents of a signed permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001726The number of visible inversions of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000829The Ulam distance of a permutation to the identity permutation. St000702The number of weak deficiencies of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St001480The number of simple summands of the module J^2/J^3. St000144The pyramid weight of the Dyck path. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000060The greater neighbor of the maximum. St000326The position of the first one in a binary word after appending a 1 at the end. St000420The number of Dyck paths that are weakly above a Dyck path. St000485The length of the longest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000925The number of topologically connected components of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000216The absolute length of a permutation. St000376The bounce deficit of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000653The last descent of a permutation. St000728The dimension of a set partition. St000809The reduced reflection length of the permutation. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001331The size of the minimal feedback vertex set. St001372The length of a longest cyclic run of ones of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001684The reduced word complexity of a permutation. St001812The biclique partition number of a graph. St001668The number of points of the poset minus the width of the poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000014The number of parking functions supported by a Dyck path. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. 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. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001658The total number of rook placements on a Ferrers board. St001820The size of the image of the pop stack sorting operator. St000335The difference of lower and upper interactions. St000444The length of the maximal rise of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001523The degree of symmetry of a Dyck path. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001933The largest multiplicity of a part in an integer partition. St001959The product of the heights of the peaks of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001955The number of natural descents for set-valued two row standard Young tableaux. St001432The order dimension of the partition. St000225Difference between largest and smallest parts in a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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 $eA$. St001863The number of weak excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St000862The number of parts of the shifted shape of a permutation. St001864The number of excedances of a signed permutation. St001946The number of descents in a parking function. St000681The Grundy value of Chomp on Ferrers diagrams. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St000993The multiplicity of the largest part of an integer partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St000937The number of positive values of the symmetric group character corresponding to the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St000815The number of semistandard Young tableaux of partition weight of given shape. St001570The minimal number of edges to add to make a graph Hamiltonian. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000761The number of ascents in an integer composition. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001566The length of the longest arithmetic progression in a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001928The number of non-overlapping descents in a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. 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. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St001060The distinguishing index of a graph. St000699The toughness times the least common multiple of 1,. St001651The Frankl number of a lattice. St000264The girth of a graph, which is not a tree. St000741The Colin de Verdière graph invariant. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000456The monochromatic index of a connected graph. St000736The last entry in the first row of a semistandard tableau. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. 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. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001948The number of augmented double ascents of a permutation. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001118The acyclic chromatic index of a graph. 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. 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.
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