Your data matches 57 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000742
Mapping
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000742: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
 => [[1]]
 => [1] => 0
[1,2] => [[1,0],[0,1]]
 => [[1,1],[2]]
 => [3,1,2] => 1
[2,1] => [[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1,3] => 2
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [6,4,5,1,2,3] => 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [5,4,6,1,2,3] => 2
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [6,3,4,1,2,5] => 2
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [4,3,5,1,2,6] => 3
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [5,2,6,1,3,4] => 3
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [4,2,5,1,3,6] => 4
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => [10,8,9,5,6,7,1,2,3,4] => 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [9,8,10,5,6,7,1,2,3,4] => 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [10,7,8,5,6,9,1,2,3,4] => 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [8,7,9,5,6,10,1,2,3,4] => 3
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [9,6,10,5,7,8,1,2,3,4] => 3
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [8,6,9,5,7,10,1,2,3,4] => 4
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [10,8,9,4,5,6,1,2,3,7] => 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [9,8,10,4,5,6,1,2,3,7] => 3
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [10,6,7,4,5,8,1,2,3,9] => 3
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [7,6,8,4,5,9,1,2,3,10] => 4
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [9,5,10,4,6,7,1,2,3,8] => 4
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [7,5,8,4,6,9,1,2,3,10] => 5
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [10,7,8,3,4,9,1,2,5,6] => 3
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [8,7,9,3,4,10,1,2,5,6] => 4
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [10,6,7,3,4,8,1,2,5,9] => 4
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [7,6,8,3,4,9,1,2,5,10] => 5
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [8,4,9,3,5,10,1,2,6,7] => 5
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [7,4,8,3,5,9,1,2,6,10] => 6
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [9,6,10,2,7,8,1,3,4,5] => 4
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [8,6,9,2,7,10,1,3,4,5] => 5
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [9,5,10,2,6,7,1,3,4,8] => 5
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [7,5,8,2,6,9,1,3,4,10] => 6
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,3],[2,3,4],[3,4],[4]]
 => [8,4,9,2,5,10,1,3,6,7] => 6
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,4],[2,3,4],[3,4],[4]]
 => [7,4,8,2,5,9,1,3,6,10] => 7
Description
The number of big ascents of a permutation after prepending zero. Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$. It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]). G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]]. [[St000646]] is the statistic without the border condition $\pi(0) = 0$.
Matching statistic: St000494
(load all 14 compositions to match this statistic)
Mapping
St000494: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => ? = 0 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 2 = 3 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 3 = 4 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 3 = 4 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => 3 = 4 - 1
[2,4,1,3] => 3 = 4 - 1
[2,4,3,1] => 4 = 5 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 3 = 4 - 1
[3,2,1,4] => 3 = 4 - 1
[3,2,4,1] => 4 = 5 - 1
[3,4,1,2] => 4 = 5 - 1
[3,4,2,1] => 5 = 6 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 4 = 5 - 1
[4,2,1,3] => 4 = 5 - 1
[4,2,3,1] => 5 = 6 - 1
[4,3,1,2] => 5 = 6 - 1
[4,3,2,1] => 6 = 7 - 1
Description
The number of inversions of distance at most 3 of a permutation. An inversion of a permutation $\pi$ is a pair $i < j$ such that $\sigma(i) > \sigma(j)$. Let $j-i$ be the distance of such an inversion. Then inversions of distance at most 1 are then exactly the descents of $\pi$, see [[St000021]]. This statistic counts the number of inversions of distance at most 3.
Matching statistic: St000794
(load all 7 compositions to match this statistic)
Mapping
St000794: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => ? = 0 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 2 = 3 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 2 = 3 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 3 = 4 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 3 = 4 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 3 = 4 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 5 = 6 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 4 = 5 - 1
[2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => 3 = 4 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 5 = 6 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 4 = 5 - 1
[3,2,1,4] => 3 = 4 - 1
[3,2,4,1] => 5 = 6 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 4 = 5 - 1
[4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => 3 = 4 - 1
[4,2,1,3] => 3 = 4 - 1
[4,2,3,1] => 4 = 5 - 1
[4,3,1,2] => 4 = 5 - 1
[4,3,2,1] => 6 = 7 - 1
Description
The mak of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns $(2\underline{31})$, $(\underline{32}1)$, $(1\underline{32})$, $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000795
(load all 4 compositions to match this statistic)
Mapping
St000795: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => ? = 0 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 3 = 4 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 2 = 3 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => 3 = 4 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 2 = 3 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => 3 = 4 - 1
[2,3,4,1] => 5 = 6 - 1
[2,4,1,3] => 4 = 5 - 1
[2,4,3,1] => 4 = 5 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 5 = 6 - 1
[3,2,1,4] => 2 = 3 - 1
[3,2,4,1] => 6 = 7 - 1
[3,4,1,2] => 4 = 5 - 1
[3,4,2,1] => 4 = 5 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 4 = 5 - 1
[4,2,1,3] => 3 = 4 - 1
[4,2,3,1] => 5 = 6 - 1
[4,3,1,2] => 3 = 4 - 1
[4,3,2,1] => 3 = 4 - 1
Description
The mad of a permutation. According to [1], this is the sum of twice the number of occurrences of the vincular pattern of $(2\underline{31})$ plus the number of occurrences of the vincular patterns $(\underline{31}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000796
(load all 6 compositions to match this statistic)
Mapping
St000796: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => ? = 0 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 2 = 3 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 3 = 4 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => 4 = 5 - 1
[1,4,3,2] => 5 = 6 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 3 = 4 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 3 = 4 - 1
[2,4,3,1] => 4 = 5 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 4 = 5 - 1
[3,2,1,4] => 3 = 4 - 1
[3,2,4,1] => 3 = 4 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 3 = 4 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 5 = 6 - 1
[4,2,1,3] => 4 = 5 - 1
[4,2,3,1] => 4 = 5 - 1
[4,3,1,2] => 5 = 6 - 1
[4,3,2,1] => 6 = 7 - 1
Description
The stat' of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns $(\underline{13}2)$, $(\underline{31}2)$, $(\underline{32}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000797
(load all 5 compositions to match this statistic)
Mapping
St000797: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => ? = 0 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 2 = 3 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 3 = 4 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 3 = 4 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => 3 = 4 - 1
[1,4,3,2] => 5 = 6 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 4 = 5 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 4 = 5 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 3 = 4 - 1
[3,2,1,4] => 3 = 4 - 1
[3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => 3 = 4 - 1
[3,4,2,1] => 4 = 5 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 5 = 6 - 1
[4,2,1,3] => 4 = 5 - 1
[4,2,3,1] => 4 = 5 - 1
[4,3,1,2] => 5 = 6 - 1
[4,3,2,1] => 6 = 7 - 1
Description
The stat`` of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns $(1\underline{32})$, $(3\underline{12})$, $(3\underline{21})$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000798
(load all 16 compositions to match this statistic)
Mapping
St000798: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => ? = 0 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 2 = 3 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 3 = 4 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 3 = 4 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => 3 = 4 - 1
[1,4,3,2] => 5 = 6 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 4 = 5 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 4 = 5 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 4 = 5 - 1
[3,2,1,4] => 3 = 4 - 1
[3,2,4,1] => 3 = 4 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 3 = 4 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 5 = 6 - 1
[4,2,1,3] => 4 = 5 - 1
[4,2,3,1] => 4 = 5 - 1
[4,3,1,2] => 5 = 6 - 1
[4,3,2,1] => 6 = 7 - 1
Description
The makl of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns $(1\underline{32})$, $(\underline{31}2)$, $(\underline{32}1)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000833
(load all 29 compositions to match this statistic)
Mapping
St000833: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => ? = 0 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 2 = 3 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 3 = 4 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 3 = 4 - 1
[2,1,3,4] => 3 = 4 - 1
[2,1,4,3] => 4 = 5 - 1
[2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 3 = 4 - 1
[3,1,2,4] => 3 = 4 - 1
[3,1,4,2] => 4 = 5 - 1
[3,2,1,4] => 5 = 6 - 1
[3,2,4,1] => 4 = 5 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 3 = 4 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 4 = 5 - 1
[4,2,1,3] => 5 = 6 - 1
[4,2,3,1] => 4 = 5 - 1
[4,3,1,2] => 5 = 6 - 1
[4,3,2,1] => 6 = 7 - 1
Description
The comajor index of a permutation. This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Matching statistic: St001583
(load all 14 compositions to match this statistic)
Mapping
St001583: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => ? = 0 - 1
[1,2] => 1 = 2 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 3 = 4 - 1
[1,3,2] => 2 = 3 - 1
[2,1,3] => 2 = 3 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 6 = 7 - 1
[1,2,4,3] => 5 = 6 - 1
[1,3,2,4] => 5 = 6 - 1
[1,3,4,2] => 4 = 5 - 1
[1,4,2,3] => 4 = 5 - 1
[1,4,3,2] => 3 = 4 - 1
[2,1,3,4] => 5 = 6 - 1
[2,1,4,3] => 4 = 5 - 1
[2,3,1,4] => 4 = 5 - 1
[2,3,4,1] => 3 = 4 - 1
[2,4,1,3] => 3 = 4 - 1
[2,4,3,1] => 2 = 3 - 1
[3,1,2,4] => 4 = 5 - 1
[3,1,4,2] => 3 = 4 - 1
[3,2,1,4] => 3 = 4 - 1
[3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 2 = 3 - 1
[4,2,1,3] => 2 = 3 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 1 = 2 - 1
[4,3,2,1] => 0 = 1 - 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St000391
(load all 2 compositions to match this statistic)
Mapping
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1] => => ? = 0 - 1
[1,2] => 0 => 0 = 1 - 1
[2,1] => 1 => 1 = 2 - 1
[1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => 01 => 2 = 3 - 1
[2,1,3] => 10 => 1 = 2 - 1
[2,3,1] => 01 => 2 = 3 - 1
[3,1,2] => 10 => 1 = 2 - 1
[3,2,1] => 11 => 3 = 4 - 1
[1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => 001 => 3 = 4 - 1
[1,3,2,4] => 010 => 2 = 3 - 1
[1,3,4,2] => 001 => 3 = 4 - 1
[1,4,2,3] => 010 => 2 = 3 - 1
[1,4,3,2] => 011 => 5 = 6 - 1
[2,1,3,4] => 100 => 1 = 2 - 1
[2,1,4,3] => 101 => 4 = 5 - 1
[2,3,1,4] => 010 => 2 = 3 - 1
[2,3,4,1] => 001 => 3 = 4 - 1
[2,4,1,3] => 010 => 2 = 3 - 1
[2,4,3,1] => 011 => 5 = 6 - 1
[3,1,2,4] => 100 => 1 = 2 - 1
[3,1,4,2] => 101 => 4 = 5 - 1
[3,2,1,4] => 110 => 3 = 4 - 1
[3,2,4,1] => 101 => 4 = 5 - 1
[3,4,1,2] => 010 => 2 = 3 - 1
[3,4,2,1] => 011 => 5 = 6 - 1
[4,1,2,3] => 100 => 1 = 2 - 1
[4,1,3,2] => 101 => 4 = 5 - 1
[4,2,1,3] => 110 => 3 = 4 - 1
[4,2,3,1] => 101 => 4 = 5 - 1
[4,3,1,2] => 110 => 3 = 4 - 1
[4,3,2,1] => 111 => 6 = 7 - 1
Description
The sum of the positions of the ones in a binary word.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000947The major index east count of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000981The length of the longest zigzag subpath. St001877Number of indecomposable injective modules with projective dimension 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000456The monochromatic index of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000454The largest eigenvalue of a graph if it is integral. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001060The distinguishing index of a graph. St001645The pebbling number of a connected 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000646The number of big ascents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000422The energy of a graph, if it is integral. St000264The girth of a graph, which is not a tree. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000259The diameter of a connected graph. 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 $eA$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001684The reduced word complexity of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001960The number of descents of a permutation minus one if its first entry is not one. St000077The number of boxed and circled entries. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.