- FindStat

Your data matches 76 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000990
(load all 64 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first ascent of a permutation. For a permutation

$\pi$ , this is the smallest index such that

$\pi(i) < \pi(i+1)$ . For the first descent, see [[St000654]].

Matching statistic: St000542
(load all 6 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left-to-right-minima of a permutation. An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a left-to-right-minimum if there does not exist a j < i such that

$\sigma_j < \sigma_i$ .

Matching statistic: St000745
(load all 33 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St000297
(load all 10 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => 1 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => 01 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => 01 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => 10 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 010 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 101 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 010 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0010 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0010 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0100 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0101 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0010 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0100 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 0100 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 0100 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1000 => 1 = 2 - 1

Description

The number of leading ones in a binary word.

Matching statistic: St000352
(load all 6 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Elizalde-Pak rank of a permutation. This is the largest

$k$ such that

$\pi(i) > k$ for all

$i\leq k$ . According to [1], the length of the longest increasing subsequence in a

$321$ -avoiding permutation is equidistributed with the rank of a

$132$ -avoiding permutation.

Matching statistic: St000541
(load all 6 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation

$\pi$ of length

$n$ , this is the number of indices

$2 \leq j \leq n$ such that for all

$1 \leq i < j$ , the pair

$(i,j)$ is an inversion of

$\pi$ .

Matching statistic: St000864
(load all 4 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of circled entries of the shifted recording tableau of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the number of circled entries in

$Q$ .

Matching statistic: St000326
(load all 6 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [2,1] => [2,1] => 1 => 1
[1,1,0,0]
 => [1,2] => [1,2] => 0 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,3,1] => 10 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => 01 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => 10 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,3,2] => 01 => 2
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,4,3,1] => 101 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,4,1,3] => 100 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 101 => 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,4,1,3] => 100 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,4,3] => 101 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => 011 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,4,3,2] => 011 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [3,1,4,2] => 110 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => 100 => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,4,2] => 110 => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,4,3,2] => 011 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,3,2] => 011 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,3,2] => 110 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,4,3,2] => 011 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,5,4,3,1] => 1011 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,5,4,1,3] => 1001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,5,1,4,3] => 1010 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,5,4,1,3] => 1001 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,5,1,4,3] => 1010 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,5,4,3] => 1011 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,5,1,4,3] => 1010 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,5,4,1,3] => 1001 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,5,1,4,3] => 1010 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,5,4,3] => 1011 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,4,3] => 1011 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,5,1,4,3] => 1010 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,5,4,3,2] => 0111 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [3,1,5,4,2] => 1101 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [3,1,5,4,2] => 1101 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,5,1,4,2] => 1100 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [3,5,4,1,2] => 1001 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,5,1,4,2] => 1100 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,5,4,2] => 1101 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,1,5,4,2] => 1101 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [3,5,1,4,2] => 1100 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,5,4,2] => 1101 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,5,4,3,2] => 0111 => 2

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St000363

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [1,2] => [2] => ([],2)
 => 1
[1,1,0,0]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [3] => ([],3)
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4] => ([],4)
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5] => ([],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => 2

Description

The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of

$G$ . This is also the number of independent sets of maximal cardinality of

$G$ .

Matching statistic: St000382

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first part of an integer composition.

The following 66 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000504The cardinality of the first block of a set partition. St000971The smallest closer of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000234The number of global ascents of a permutation. St000734The last entry in the first row of a standard tableau. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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

$c_{n-1}$ . St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001271The competition number of a graph. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000068The number of minimal elements in a poset. St000390The number of runs of ones in a binary word. St000883The number of longest increasing subsequences of a permutation. St000237The number of small exceedances. St000260The radius of a connected graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000007The number of saliances of the permutation. St000678The number of up steps after the last double rise of a Dyck path. St000054The first entry of the permutation. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St000654The first descent of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000740The last entry of a permutation. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. 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. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001498The normalised height of a Nakayama algebra with magnitude 1. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000259The diameter of a connected graph. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000663The number of right floats of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001552The number of inversions between excedances and fixed points of a permutation. St001728The number of invisible descents of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001545The second Elser number of a connected graph. St001948The number of augmented double ascents of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000221The number of strong fixed points of a permutation. St000461The rix statistic of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000455The second largest eigenvalue of a graph if it is integral. 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

$eA$ . St001904The length of the initial strictly increasing segment of a parking function. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ .