- FindStat

Your data matches 236 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001663
(load all 66 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001663: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the Hertzsprung pattern 132 in a permutation. A Hertzsprung occurrence of the pattern

$\tau=(\tau_1,\dots,\tau_k)$ in a permutation

$\pi$ is a factor

$\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of

$\pi$ such that

$\pi_{i+j-1} - \tau_j$ is constant for

$1\leq j\leq k$ .

Matching statistic: St001141
(load all 39 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 0

Description

The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.

Matching statistic: St001137
(load all 3 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of simple modules that are 3-regular in the corresponding Nakayama algebra.

Matching statistic: St001130
(load all 2 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001130: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of two successive successions in a permutation.

Matching statistic: St000929
(load all 21 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 90%●distinct values known / distinct values provided: 67%

Values

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

Description

The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is

$0$ for partitions

$\lambda \neq 1^n$ and

$1$ for

$\lambda = 1^n$ .

Matching statistic: St000921

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000921: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => []
 => ?
 => ? => ? = 0
[1,0,1,0]
 => [1]
 => []
 => ? => ? ∊ {0,0}
[1,1,0,0]
 => []
 => ?
 => ? => ? ∊ {0,0}
[1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 1 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 1 => 0
[1,1,0,0,1,0]
 => [2]
 => []
 => ? => ? ∊ {0,0,1}
[1,1,0,1,0,0]
 => [1]
 => []
 => ? => ? ∊ {0,0,1}
[1,1,1,0,0,0]
 => []
 => ?
 => ? => ? ∊ {0,0,1}
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 01 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 01 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 11 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 11 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 11 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 0 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 0 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 1 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 1 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 1 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => []
 => ? => ? ∊ {0,0,1,1}
[1,1,1,0,0,1,0,0]
 => [2]
 => []
 => ? => ? ∊ {0,0,1,1}
[1,1,1,0,1,0,0,0]
 => [1]
 => []
 => ? => ? ∊ {0,0,1,1}
[1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? => ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => 101 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 101 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => 001 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 001 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 001 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => 111 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 111 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => 011 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 011 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 011 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 111 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 111 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 111 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 111 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 10 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 10 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 00 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 00 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 00 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 11 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 11 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 01 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 01 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 01 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 11 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 11 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 11 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => 11 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 1 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => 1 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 0 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 0 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => 0 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => 1 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => 1 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => 1 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => 1 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 => ? => ? ∊ {1,1,1,1,1}
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => ? => ? ∊ {1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => ? => ? ∊ {1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 => ? => ? ∊ {1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? => ? ∊ {1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [4,3,2,1]
 => 0101 => 0
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 => ? => ? ∊ {0,0,0,0,0,2}
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 => ? => ? ∊ {0,0,0,0,0,2}
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 => ? => ? ∊ {0,0,0,0,0,2}
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 => ? => ? ∊ {0,0,0,0,0,2}
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => []
 => ? => ? ∊ {0,0,0,0,0,2}
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? => ? ∊ {0,0,0,0,0,2}

Description

The number of internal inversions of a binary word. Let

$\bar w$ be the non-decreasing rearrangement of

$w$ , that is,

$\bar w$ is sorted. An internal inversion is a pair

$i < j$ such that

$w_i > w_j$ and

$\bar w_i = \bar w_j$ . For example, the word

$110$ has two inversions, but only the second is internal.

Matching statistic: St001175
(load all 8 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%

Values

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

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St001498
(load all 47 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 88%●distinct values known / distinct values provided: 33%

Values

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

Description

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.

Matching statistic: St001292
(load all 5 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 84%●distinct values known / distinct values provided: 33%

Values

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

Description

The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Here

$A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].

Matching statistic: St001695
(load all 2 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 84%●distinct values known / distinct values provided: 33%

Values

[1,0]
 => []
 => ?
 => ?
 => ? = 0
[1,0,1,0]
 => [1]
 => []
 => []
 => ? ∊ {0,0}
[1,1,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0}
[1,0,1,0,1,0]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,0,0,1,0]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1}
[1,1,0,1,0,0]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[1,1,1,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,0,0,1,0,0]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,0,1,0,0,0]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,1,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => [[1,2,3],[4]]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => [[1,2,3],[4]]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => [[1,2,3]]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => [[1,2,3]]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => [[1,2]]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => [[1,2]]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => [[1,2]]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => [[1]]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1}
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2}

Description

The natural comajor index of a standard Young tableau. A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation. The natural comajor index of a tableau of size

$n$ with natural descent set

$D$ is then

$\sum_{d\in D} n-d$ .

The following 226 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000234The number of global ascents of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001730The number of times the path corresponding to a binary word crosses the base line. St000567The sum of the products of all pairs of parts. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000296The length of the symmetric border of a binary word. St000377The dinv defect of an integer partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000629The defect of a binary word. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

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

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000666The number of right tethers of a permutation. St000661The number of rises of length 3 of a Dyck path. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000379The number of Hamiltonian cycles in a graph. St000516The number of stretching pairs of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. 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. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000022The number of fixed points of a permutation. St001586The number of odd parts smaller than the largest even part in an integer partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000931The number of occurrences of the pattern UUU in a Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. St001162The minimum jump of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000664The number of right ropes of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000260The radius of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000221The number of strong fixed points of a permutation. St000731The number of double exceedences of a permutation. St000989The number of final rises of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001810The number of fixed points of a permutation smaller than its largest moved point. St000028The number of stack-sorts needed to sort a permutation. St000056The decomposition (or block) number of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001461The number of topologically connected components of the chord diagram of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001330The hat guessing number of a graph. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001875The number of simple modules with projective dimension at most 1. St000627The exponent of a binary word. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001939The number of parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000181The number of connected components of the Hasse diagram for the poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001846The number of elements which do not have a complement in the lattice. 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

$eA$ . St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001720The minimal length of a chain of small intervals in a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000058The order of a permutation. St001820The size of the image of the pop stack sorting operator. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001868The number of alignments of type NE of a signed permutation. St000741The Colin de Verdière graph invariant. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St000488The number of cycles of a permutation of length at most 2. St001394The genus of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. 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

$g$ . St001344The neighbouring number of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St000764The number of strong records in an integer composition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000883The number of longest increasing subsequences of a permutation. St000352The Elizalde-Pak rank of a permutation. St000454The largest eigenvalue of a graph if it is integral.