- FindStat

Your data matches 328 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000374
(load all 18 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St001142
(load all 95 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001505
(load all 96 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St001505: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1

Description

The number of elements generated by the Dyck path as a map in the full transformation monoid. We view the resolution quiver of a Dyck path (corresponding to an LNakayamaalgebra) as a transformation and associate to it the submonoid generated by this map in the full transformation monoid.

Matching statistic: St001461
(load all 2 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => {{1,2}}
 => [2,1] => 1
[1,0,1,0]
 => [3,1,2] => {{1,3},{2}}
 => [3,2,1] => 2
[1,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => [2,3,1] => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => [4,2,3,1] => 3
[1,0,1,1,0,0]
 => [3,1,4,2] => {{1,3,4},{2}}
 => [3,2,4,1] => 2
[1,1,0,0,1,0]
 => [2,4,1,3] => {{1,2,4},{3}}
 => [2,4,3,1] => 2
[1,1,0,1,0,0]
 => [4,3,1,2] => {{1,4},{2,3}}
 => [4,3,2,1] => 2
[1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => 4
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => {{1,4,5},{2},{3}}
 => [4,2,3,5,1] => 3
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => {{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 3
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => {{1,5},{2},{3,4}}
 => [5,2,4,3,1] => 3
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => [3,2,4,5,1] => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 3
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => {{1,5},{2,3},{4}}
 => [5,3,2,4,1] => 3
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => {{1,5},{2,4},{3}}
 => [5,4,3,2,1] => 3
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => {{1,4,5},{2,3}}
 => [4,3,2,5,1] => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => [2,3,5,4,1] => 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => {{1,5},{2,3,4}}
 => [5,3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}}
 => [5,2,3,4,6,1] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}}
 => [4,2,3,6,5,1] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}}
 => [6,2,3,5,4,1] => 4
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
 => [4,2,3,5,6,1] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}}
 => [3,2,6,4,5,1] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
 => [3,2,5,4,6,1] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}}
 => [6,2,4,3,5,1] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}}
 => [6,2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
 => [5,2,4,3,6,1] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
 => [3,2,6,5,4,1] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}}
 => [6,2,4,5,3,1] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
 => [3,2,4,5,6,1] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}}
 => [2,6,3,4,5,1] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}}
 => [2,5,3,4,6,1] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}}
 => [2,4,3,5,6,1] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}}
 => [6,3,2,4,5,1] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}}
 => [5,3,2,4,6,1] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}}
 => [6,4,3,2,5,1] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}}
 => [5,6,3,4,1,2] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}}
 => [5,4,3,2,6,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}}
 => [4,3,2,6,5,1] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}}
 => [6,3,2,5,4,1] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}}
 => [6,4,3,5,2,1] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => {{1,4,5,6},{2,3}}
 => [4,3,2,5,6,1] => 2

Description

The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation

$\pi\in\mathfrak S_n$ is obtained by placing labels

$1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from

$i$ to

$\pi(i)$ for every label

$i$ . This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation

$\pi\in\mathfrak S_n$ stabilizes an interval

$I=\{a,a+1,\dots,b\}$ if

$\pi(I)=I$ . It is stabilized-interval-free, if the only interval

$\pi$ stablizes is

$\{1,\dots,n\}$ . Thus, this statistic is

$1$ if

$\pi$ is stabilized-interval-free.

Matching statistic: St001462
(load all 2 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [2,1] => [[1],[2]]
 => 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => [[1,3],[2]]
 => 2
[1,1,0,0]
 => [2,3,1] => [3,2,1] => [[1],[2],[3]]
 => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [[1,3,4],[2]]
 => 3
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => [[1,4],[2],[3]]
 => 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
 => 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => [[1,3,4,5],[2]]
 => 4
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
 => 3
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
 => 3
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [4,1,2,5,3] => [[1,3,4],[2,5]]
 => 3
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => [[1,4,5],[2],[3]]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => [[1,5],[2],[3],[4]]
 => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,5,2,4] => [[1,3,5],[2,4]]
 => 3
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,1,5,2,3] => [[1,3,5],[2,4]]
 => 3
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [3,1,5,4,2] => [[1,3],[2,4],[5]]
 => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => [[1,5],[2],[3],[4]]
 => 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,2,1,5,3] => [[1,4],[2,5],[3]]
 => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [4,3,1,5,2] => [[1,4],[2,5],[3]]
 => 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => [[1,3,4,5,6],[2]]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,5,1,2,3,4] => [[1,4,5,6],[2],[3]]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,4,1,2,3,5] => [[1,4,5,6],[2],[3]]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [5,1,2,3,6,4] => [[1,3,4,5],[2,6]]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,5,4,1,2,3] => [[1,5,6],[2],[3],[4]]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,3,1,2,4,5] => [[1,4,5,6],[2],[3]]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,5,3,1,2,4] => [[1,5,6],[2],[3],[4]]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,1,2,6,3,5] => [[1,3,4,6],[2,5]]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,1,2,6,3,4] => [[1,3,4,6],[2,5]]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [4,1,2,6,5,3] => [[1,3,4],[2,5],[6]]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,4,3,1,2,5] => [[1,5,6],[2],[3],[4]]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,3,1,2,6,4] => [[1,4,5],[2,6],[3]]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [5,4,1,2,6,3] => [[1,4,5],[2,6],[3]]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,2,1,3,4,5] => [[1,4,5,6],[2],[3]]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,5,2,1,3,4] => [[1,5,6],[2],[3],[4]]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,4,2,1,3,5] => [[1,5,6],[2],[3],[4]]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [5,2,1,3,6,4] => [[1,4,5],[2,6],[3]]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,5,4,2,1,3] => [[1,6],[2],[3],[4],[5]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [3,1,6,2,4,5] => [[1,3,5,6],[2,4]]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [3,1,6,5,2,4] => [[1,3,6],[2,4],[5]]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,1,6,2,3,5] => [[1,3,5,6],[2,4]]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [6,1,5,2,3,4] => [[1,3,5,6],[2],[4]]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,1,6,5,2,3] => [[1,3,6],[2,4],[5]]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,1,6,4,2,5] => [[1,3,6],[2,4],[5]]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [3,1,5,2,6,4] => [[1,3,5],[2,4,6]]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,4,1,6,2,3] => [[1,4,6],[2,5],[3]]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,1,6,5,4,2] => [[1,3],[2,4],[5],[6]]
 => 2

Description

The number of factors of a standard tableaux under concatenation. The concatenation of two standard Young tableaux

$T_1$ and

$T_2$ is obtained by adding the largest entry of

$T_1$ to each entry of

$T_2$ , and then appending the rows of the result to

$T_1$ , see [1, dfn 2.10]. This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.

Matching statistic: St000996
(load all 10 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

Matching statistic: St000925
(load all 6 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => {{1,2}}
 => 1
[1,0,1,0]
 => [3,1,2] => {{1,3},{2}}
 => 2
[1,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 3
[1,0,1,1,0,0]
 => [3,1,4,2] => {{1,3,4},{2}}
 => 2
[1,1,0,0,1,0]
 => [2,4,1,3] => {{1,2,4},{3}}
 => 2
[1,1,0,1,0,0]
 => [4,3,1,2] => {{1,4},{2,3}}
 => 2
[1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
 => 4
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => {{1,4,5},{2},{3}}
 => 3
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => {{1,3,5},{2},{4}}
 => 3
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => {{1,5},{2},{3,4}}
 => 3
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => 3
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => {{1,5},{2,3},{4}}
 => 3
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => {{1,5},{2,4},{3}}
 => 3
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => {{1,4,5},{2,3}}
 => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => {{1,2,5},{3,4}}
 => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => {{1,5},{2,3,4}}
 => 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}}
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}}
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}}
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}}
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}}
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}}
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}}
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}}
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}}
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}}
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}}
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}}
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}}
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}}
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}}
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}}
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}}
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}}
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}}
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}}
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => {{1,4,5,6},{2,3}}
 => 2
[]
 => [1] => {{1}}
 => ? = 1

Description

The number of topologically connected components of a set partition. For example, the set partition

$\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components

$\{1,4,5,6\}$ and

$\{2,3\}$ . The number of set partitions with only one block is [[oeis:A099947]].

Matching statistic: St000172
(load all 2 compositions to match this statistic)

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The Grundy number of a graph. The Grundy number

$\Gamma(G)$ is defined to be the largest

$k$ such that

$G$ admits a greedy

$k$ -coloring. Any order of the vertices of

$G$ induces a greedy coloring by assigning to the

$i$ -th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that

$\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$ , where

$\chi(G)$ is the chromatic number of

$G$ ([[St000098]]), and where

$\Delta(G)$ is the maximal degree of a vertex of

$G$ ([[St000171]]).

Matching statistic: St000213
(load all 10 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of weak exceedances (also weak excedences) of a permutation. This is defined as

$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$ The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of

$\sigma$ .

Matching statistic: St001581
(load all 3 compositions to match this statistic)

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

The following 318 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000288The number of ones in a binary word. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001866The nesting alignments of a signed permutation. St001896The number of right descents of a signed permutations. St001330The hat guessing number of a graph. St000507The number of ascents of a standard tableau. St001820The size of the image of the pop stack sorting operator. St001487The number of inner corners of a skew partition. St001720The minimal length of a chain of small intervals in a lattice. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001645The pebbling number of a connected graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000647The number of big descents of a permutation. St000884The number of isolated descents of a permutation. St000703The number of deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000007The number of saliances of the permutation. St000542The number of left-to-right-minima of a permutation. St000292The number of ascents of a binary word. St000354The number of recoils of a permutation. St000451The length of the longest pattern of the form k 1 2. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000015The number of peaks of a Dyck path. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000201The number of leaf nodes in a binary tree. St000214The number of adjacencies of a permutation. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000443The number of long tunnels of a Dyck path. St000746The number of pairs with odd minimum in a perfect matching. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001589The nesting number of a perfect matching. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000053The number of valleys of the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000236The number of cyclical small weak excedances. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000710The number of big deficiencies of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001152The number of pairs with even minimum in a perfect matching. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001874Lusztig's a-function for the symmetric group. St001964The interval resolution global dimension of a poset. St001712The number of natural descents of a standard Young tableau. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001668The number of points of the poset minus the width of the poset. St000097The order of the largest clique of the graph. St000386The number of factors DDU in a Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000662The staircase size of the code of a permutation. 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

$eA$ . 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

$eA$ . St001877Number of indecomposable injective modules with projective dimension 2. St001435The number of missing boxes in the first row. St000670The reversal length of a permutation. St000352The Elizalde-Pak rank of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000237The number of small exceedances. St000919The number of maximal left branches of a binary tree. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001083The number of boxed occurrences of 132 in a permutation. St001394The genus of a permutation. St000568The hook number of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000098The chromatic number of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000306The bounce count of a Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000366The number of double descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001115The number of even descents of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000071The number of maximal chains in a poset. St000356The number of occurrences of the pattern 13-2. St000850The number of 1/2-balanced pairs in a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001530The depth of a Dyck path. St001665The number of pure excedances of a permutation. St000264The girth of a graph, which is not a tree. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001060The distinguishing index of a graph. 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. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001728The number of invisible descents of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. 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. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001061The number of indices that are both descents and recoils of a permutation. St001737The number of descents of type 2 in a permutation. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001840The number of descents of a set partition. St000056The decomposition (or block) number of a permutation. St000238The number of indices that are not small weak excedances. St000254The nesting number of a set partition. St000272The treewidth of a graph. St000307The number of rowmotion orbits of a poset. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000831The number of indices that are either descents or recoils. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000999Number of indecomposable projective module with injective 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. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001220The width of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001405The number of bonds in a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. 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. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001928The number of non-overlapping descents in a permutation. St000004The major index of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000080The rank of the poset. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000504The cardinality of the first block of a set partition. St000640The rank of the largest boolean interval in a poset. St000650The number of 3-rises of a permutation. St000653The last descent of a permutation. St000779The tier of a permutation. St000794The mak of a permutation. St000822The Hadwiger number of the graph. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St000918The 2-limited packing number of a graph. St000961The shifted major index of a permutation. St000989The number of final rises of a permutation. St001029The size of the core of a graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001062The maximal size of a block of a set partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001116The game chromatic number of a graph. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001202Call 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. 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: St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001388The number of non-attacking neighbors of a permutation. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001580The acyclic chromatic number of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001670The connected partition number of a graph. St001726The number of visible inversions of a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001902The number of potential covers of a poset. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000741The Colin de Verdière graph invariant. St001875The number of simple modules with projective dimension at most 1. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation.