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Your data matches 373 different statistics following compositions of up to 3 maps.
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Matching statistic: St000255
(load all 21 compositions to match this statistic)

Mapping

St000255: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 5
[2,1,3,4] => 1
[2,1,4,3] => 3
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1

Description

The number of reduced Kogan faces with the permutation as type. This is equivalent to finding the number of ways to represent the permutation

$\pi \in S_{n+1}$ as a reduced subword of

$s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$ , or the number of reduced pipe dreams for

$\pi$ .

Matching statistic: St000427
(load all 5 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000427: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,3,2] => 0 = 1 - 1
[2,1,3] => [2,1,3] => 0 = 1 - 1
[2,3,1] => [3,2,1] => 0 = 1 - 1
[3,1,2] => [3,2,1] => 0 = 1 - 1
[3,2,1] => [3,2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => [1,4,3,2] => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => [3,2,1,4] => 0 = 1 - 1
[2,3,4,1] => [4,2,3,1] => 1 = 2 - 1
[2,4,1,3] => [3,4,1,2] => 2 = 3 - 1
[2,4,3,1] => [4,3,2,1] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => 0 = 1 - 1
[3,1,4,2] => [4,2,3,1] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => [4,2,3,1] => 1 = 2 - 1
[3,4,1,2] => [4,3,2,1] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => 1 = 2 - 1
[4,1,3,2] => [4,2,3,1] => 1 = 2 - 1
[4,2,1,3] => [4,3,2,1] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => 0 = 1 - 1

Description

The number of occurrences of the pattern 123 or of the pattern 231 in a permutation.

Matching statistic: St000430
(load all 7 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000430: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 1 = 2 - 1
[1,3,2] => [1,3,2] => 0 = 1 - 1
[2,1,3] => [2,1,3] => 0 = 1 - 1
[2,3,1] => [3,2,1] => 0 = 1 - 1
[3,1,2] => [3,2,1] => 0 = 1 - 1
[3,2,1] => [3,2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => [1,4,3,2] => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => [3,2,1,4] => 0 = 1 - 1
[2,3,4,1] => [4,2,3,1] => 1 = 2 - 1
[2,4,1,3] => [3,4,1,2] => 2 = 3 - 1
[2,4,3,1] => [4,3,2,1] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => 0 = 1 - 1
[3,1,4,2] => [4,2,3,1] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => [4,2,3,1] => 1 = 2 - 1
[3,4,1,2] => [4,3,2,1] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => 1 = 2 - 1
[4,1,3,2] => [4,2,3,1] => 1 = 2 - 1
[4,2,1,3] => [4,3,2,1] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => 0 = 1 - 1

Description

The number of occurrences of the pattern 123 or of the pattern 312 in a permutation.

Matching statistic: St001770

Mapping

Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001770: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [-1] => 1
[1,2] => [1,2] => [-1,-2] => 1
[2,1] => [2,1] => [-2,-1] => 1
[1,2,3] => [1,2,3] => [-1,-2,-3] => 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => 2
[3,1,2] => [3,1,2] => [-3,-1,-2] => 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => 1
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 1
[1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 2
[1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 1
[2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => 5
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 2
[2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 3
[3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => 1
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 2
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 1
[3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 3
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 3
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 3
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => 1
[4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 1
[4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 1
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 2
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 1

Description

The number of facets of a certain subword complex associated with the signed permutation. Let

$Q=[1,\dots,n,1,\dots,n,\dots,1,\dots,n]$ be the word of length

$n^2$ , and let

$\pi$ be a signed permutation. Then this statistic yields the number of facets of the subword complex

$\Delta(Q, \pi)$ .

Matching statistic: St001941
(load all 4 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001941: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 2
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [3,2,1] => [1,2,3] => 1
[3,1,2] => [3,2,1] => [1,2,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => [3,1,4,2] => 1
[1,2,4,3] => [1,2,4,3] => [3,2,1,4] => 2
[1,3,2,4] => [1,3,2,4] => [4,3,2,1] => 5
[1,3,4,2] => [1,4,3,2] => [2,4,3,1] => 2
[1,4,2,3] => [1,4,3,2] => [2,4,3,1] => 2
[1,4,3,2] => [1,4,3,2] => [2,4,3,1] => 2
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 2
[2,3,1,4] => [3,2,1,4] => [4,3,1,2] => 3
[2,3,4,1] => [4,2,3,1] => [1,2,3,4] => 1
[2,4,1,3] => [3,4,1,2] => [1,4,2,3] => 1
[2,4,3,1] => [4,3,2,1] => [2,4,1,3] => 1
[3,1,2,4] => [3,2,1,4] => [4,3,1,2] => 3
[3,1,4,2] => [4,2,3,1] => [1,2,3,4] => 1
[3,2,1,4] => [3,2,1,4] => [4,3,1,2] => 3
[3,2,4,1] => [4,2,3,1] => [1,2,3,4] => 1
[3,4,1,2] => [4,3,2,1] => [2,4,1,3] => 1
[3,4,2,1] => [4,3,2,1] => [2,4,1,3] => 1
[4,1,2,3] => [4,2,3,1] => [1,2,3,4] => 1
[4,1,3,2] => [4,2,3,1] => [1,2,3,4] => 1
[4,2,1,3] => [4,3,2,1] => [2,4,1,3] => 1
[4,2,3,1] => [4,3,2,1] => [2,4,1,3] => 1
[4,3,1,2] => [4,3,2,1] => [2,4,1,3] => 1
[4,3,2,1] => [4,3,2,1] => [2,4,1,3] => 1

Description

The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5.3]) with parameters given by the identity and the permutation. Also the number of paths in the Bruhat order from the identity to the permutation that are increasing with respect to a given reflection ordering as defined in Björner and Brenti [1, Section 5.2].

Matching statistic: St000119
(load all 9 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000119: 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] => [3,2,1] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [3,1,2] => 0 = 1 - 1
[2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [3,2,1,4] => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [4,3,2,1] => 4 = 5 - 1
[1,3,4,2] => [1,4,3,2] => [2,4,3,1] => 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [2,4,3,1] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [2,4,3,1] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 1 = 2 - 1
[2,3,1,4] => [3,2,1,4] => [4,3,1,2] => 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [3,4,1,2] => [1,4,2,3] => 0 = 1 - 1
[2,4,3,1] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => [4,3,1,2] => 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [4,3,1,2] => 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,1,2] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[4,2,1,3] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1

Description

The number of occurrences of the pattern 321 in a permutation.

Matching statistic: St000431

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000431: 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] => [2,1] => 0 = 1 - 1
[2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [3,2,1] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0 = 1 - 1
[2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[2,4,1,3] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[3,1,4,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[3,2,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,3,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1

Description

The number of occurrences of the pattern 213 or of the pattern 321 in a permutation.

Matching statistic: St000433
(load all 4 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000433: 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] => [2,1] => 0 = 1 - 1
[2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [3,2,1] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4 = 5 - 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0 = 1 - 1
[2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[2,4,1,3] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[3,1,4,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[3,2,4,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,1,3,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1

Description

The number of occurrences of the pattern 132 or of the pattern 321 in a permutation.

Matching statistic: St001411
(load all 9 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001411: 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] => [3,2,1] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [3,1,2] => 0 = 1 - 1
[2,3,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [3,2,1,4] => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [4,3,2,1] => 4 = 5 - 1
[1,3,4,2] => [1,4,3,2] => [2,4,3,1] => 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [2,4,3,1] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [2,4,3,1] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 1 = 2 - 1
[2,3,1,4] => [3,2,1,4] => [4,3,1,2] => 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [3,4,1,2] => [1,4,2,3] => 0 = 1 - 1
[2,4,3,1] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[3,1,2,4] => [3,2,1,4] => [4,3,1,2] => 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [4,3,1,2] => 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[3,4,1,2] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[3,4,2,1] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[4,1,2,3] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [4,2,3,1] => [1,2,3,4] => 0 = 1 - 1
[4,2,1,3] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[4,2,3,1] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[4,3,1,2] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [2,4,1,3] => 0 = 1 - 1

Description

The number of patterns 321 or 3412 in a permutation. A permutation is '''boolean''' if its principal order ideal in the (strong) Bruhat order is boolean. It is shown in [1, Theorem 5.3] that a permutation is boolean if and only if it avoids the two patterns 321 and 3412.

Matching statistic: St000078
(load all 9 compositions to match this statistic)

Mapping

Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00245: Signed permutations —standardize⟶ Permutations
St000078: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [-2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [1,-3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [-2,1,3] => [3,1,2] => 1
[2,3,1] => [2,3,1] => [3,-2,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [-2,-3,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,-4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => [1,-3,2,4] => [1,4,2,3] => 3
[1,3,4,2] => [1,3,4,2] => [1,4,-3,2] => [1,3,4,2] => 3
[1,4,2,3] => [1,4,2,3] => [-4,1,2,3] => [4,1,2,3] => 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => [1,4,3,2] => 5
[2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => [4,1,2,3] => 1
[2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => [4,1,3,2] => 2
[2,3,1,4] => [2,3,1,4] => [3,-2,1,4] => [2,4,1,3] => 2
[2,3,4,1] => [2,3,4,1] => [3,4,-2,1] => [2,3,4,1] => 1
[2,4,1,3] => [2,4,1,3] => [-2,4,1,3] => [4,3,1,2] => 1
[2,4,3,1] => [2,4,3,1] => [3,-2,-4,1] => [2,4,3,1] => 2
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [3,1,4,2] => [-4,1,-3,2] => [3,1,4,2] => 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => [4,3,1,2] => 1
[3,2,4,1] => [3,2,4,1] => [-2,4,-3,1] => [4,2,3,1] => 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 1
[3,4,2,1] => [3,4,2,1] => [2,-4,-3,1] => [2,3,4,1] => 1
[4,1,2,3] => [4,1,2,3] => [1,4,2,3] => [1,4,2,3] => 3
[4,1,3,2] => [4,1,3,2] => [3,1,-4,2] => [3,1,4,2] => 2
[4,2,1,3] => [4,2,1,3] => [-4,-2,1,3] => [3,4,1,2] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => [2,3,4,1] => 1
[4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => [-2,-3,-4,1] => [4,3,2,1] => 1

Description

The number of alternating sign matrices whose left key is the permutation. The left key of an alternating sign matrix was defined by Lascoux in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.

The following 363 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000002The number of occurrences of the pattern 123 in a permutation. St000463The number of admissible inversions of a permutation. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. 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. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000961The shifted major index of a permutation. St001438The number of missing boxes of a skew partition. St001734The lettericity of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000667The greatest common divisor of the parts of the partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001845The number of join irreducibles minus the rank of a lattice. St000456The monochromatic index of a connected graph. St001527The cyclic permutation representation number of an integer partition. St000010The length of the partition. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000048The multinomial of the parts of a partition. St000160The multiplicity of the smallest part of a partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000297The number of leading ones in a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000529The number of permutations whose descent word is the given binary word. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000675The number of centered multitunnels of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000935The number of ordered refinements of an integer partition. St000983The length of the longest alternating subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001389The number of partitions of the same length below the given integer partition. St001415The length of the longest palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001571The Cartan determinant of the integer partition. St001597The Frobenius rank of a skew partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. 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. St001614The cyclic permutation representation number of a skew partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000993The multiplicity of the largest part of an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001432The order dimension of the partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001846The number of elements which do not have a complement in the lattice. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St001128The exponens consonantiae of 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000422The energy of a graph, if it is integral. St000527The width of the poset. St000618The number of self-evacuating tableaux of given shape. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001280The number of parts of an integer partition that are at least two. St001378The product of the cohook lengths of the integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001118The acyclic chromatic index of a graph. St000259The diameter of a connected graph. St000100The number of linear extensions of a poset. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001890The maximum magnitude of the Möbius function of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St000741The Colin de Verdière graph invariant. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000420The number of Dyck paths that are weakly above a Dyck path. St000678The number of up steps after the last double rise 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. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001645The pebbling number of a connected graph. St001808The box weight or horizontal decoration of a Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001722The number of minimal chains with small intervals between a binary word and the top element. St001763The Hurwitz number of an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001943The sum of the squares of the hook lengths of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000762The sum of the positions of the weak records of an integer composition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation. St000842The breadth of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001857The number of edges in the reduced word graph of a signed permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. 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. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000562The number of internal points of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000779The tier of a permutation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. 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. 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. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001573The minimal number of edges to remove to make a graph triangle-free. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001871The number of triconnected components of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001060The distinguishing index of a graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St001435The number of missing boxes in the first row. St001624The breadth of a lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St000264The girth of a graph, which is not a tree. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. 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. St000455The second largest eigenvalue of a graph if it is integral. St000102The charge of a semistandard tableau. St000309The number of vertices with even degree. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001828The Euler characteristic of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000822The Hadwiger number of the graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001625The Möbius invariant of a lattice. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001783The number of odd automorphisms of a graph. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001738The minimal order of a graph which is not an induced subgraph of the given graph.