- FindStat

Your data matches 382 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001257
(load all 21 compositions to match this statistic)

Mapping

St001257: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 2 = 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 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1

Description

The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.

Matching statistic: St000338
(load all 20 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000338: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of pixed points of a permutation. For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{pix} \, \sigma = \textrm{length} (p)$ .

Matching statistic: St000461
(load all 13 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rix statistic of a permutation. This statistic is defined recursively as follows:

$rix([]) = 0$ , and if

$w_i = \max\{w_1, w_2,\dots, w_k\}$ , then

$rix(w) := 0$ if

$i = 1 < k$ ,

$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if

$i = k$ and

$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if

$1 < i < k$ .

Matching statistic: St001271
(load all 14 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The competition number of a graph. The competition graph of a digraph

$D$ is a (simple undirected) graph which has the same vertex set as

$D$ and has an edge between

$x$ and

$y$ if and only if there exists a vertex

$v$ in

$D$ such that

$(x, v)$ and

$(y, v)$ are arcs of

$D$ . For any graph,

$G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number

$k(G)$ is the smallest number of such isolated vertices.

Matching statistic: St001276
(load all 6 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001276: 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]
 => [2,1] => [1,1,0,0]
 => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 0

Description

The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module

$M$ over an algebra

$A$ k-regular in case it has projective dimension k and

$Ext_A^i(M,A)=0$ for

$i \neq k$ and

$Ext_A^k(M,A)$ is 1-dimensional. The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .

Matching statistic: St000908
(load all 3 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the shortest maximal antichain in a poset.

Matching statistic: St001316
(load all 8 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The domatic number of a graph. This is the maximal size of a partition of the vertices into dominating sets.

Matching statistic: St000237
(load all 5 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of small exceedances. This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Matching statistic: St000310
(load all 3 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000310: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal degree of a vertex of a graph.

Matching statistic: St000352
(load all 3 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

$k$ such that

$\pi(i) > k$ for all

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

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

$132$ -avoiding permutation.

The following 372 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St000674The number of hills of a Dyck path. St000753The Grundy value for the game of Kayles on a binary word. St000754The Grundy value for the game of removing nestings in a perfect matching. St000873The aix statistic of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001395The number of strictly unfriendly partitions of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001691The number of kings in a graph. St001728The number of invisible descents of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000273The domination number of a graph. St000335The difference of lower and upper interactions. St000654The first descent of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001481The minimal height of a peak of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000007The number of saliances of the permutation. St000990The first ascent of a permutation. St000297The number of leading ones in a binary word. St000989The number of final rises of a permutation. St001948The number of augmented double ascents of a permutation. St000906The length of the shortest maximal chain in a poset. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000257The number of distinct parts of a partition that occur at least twice. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000256The number of parts from which one can substract 2 and still get an integer partition. St000661The number of rises of length 3 of a Dyck path. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000658The number of rises of length 2 of a Dyck path. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000260The radius of a connected graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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. St000628The balance of a binary word. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. St001252Half the sum of the even parts of a partition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001524The degree of symmetry of a binary word. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000733The row containing the largest entry of a standard tableau. St000137The Grundy value of an integer partition. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001525The number of symmetric hooks on the diagonal of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001061The number of indices that are both descents and recoils of a permutation. St000941The number of characters of the symmetric group whose value on the partition is even. St001593This is the number of standard Young tableaux of the given shifted shape. St000379The number of Hamiltonian cycles in a graph. St000659The number of rises of length at least 2 of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001964The interval resolution global dimension of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001820The size of the image of the pop stack sorting operator. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000944The 3-degree of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000292The number of ascents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000348The non-inversion sum of a binary word. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000547The number of even non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000682The Grundy value of Welter's game on a binary word. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to 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. St000783The side length of the largest staircase partition fitting into a partition. St000897The number of different multiplicities of parts of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. 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. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001484The number of singletons of an integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001651The Frankl number of a lattice. St001695The natural comajor index of a standard Young tableau. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001730The number of times the path corresponding to a binary word crosses the base line. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001910The height of the middle non-run of a Dyck path. St001961The sum of the greatest common divisors of all pairs of parts. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000054The first entry of the permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000741The Colin de Verdière graph invariant. St001960The number of descents of a permutation minus one if its first entry is not one. St000764The number of strong records in an integer composition. St001520The number of strict 3-descents. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001895The oddness of a signed permutation. St001399The distinguishing number of a poset. 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. St001175The size of a partition minus the hook length of the base cell. St001846The number of elements which do not have a complement in the lattice. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index of a connected graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001557The number of inversions of the second entry of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001621The number of atoms of a lattice. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000100The number of linear extensions of a poset. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. 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. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000669The number of permutations obtained by switching ascents or descents of size 2. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. 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. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001330The hat guessing number of a graph. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000366The number of double descents of a permutation. St000534The number of 2-rises of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000618The number of self-evacuating tableaux of given shape. St000731The number of double exceedences of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001541The Gini index of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000883The number of longest increasing subsequences of a permutation. St000842The breadth of a permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000284The Plancherel distribution on integer partitions. St000441The number of successions of a permutation. St000567The sum of the products of all pairs of parts. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001128The exponens consonantiae of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001933The largest multiplicity of a part in an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001151The number of blocks with odd minimum. St000090The variation of a composition. St000133The "bounce" of a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000241The number of cyclical small excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000552The number of cut vertices of a graph. St000563The number of overlapping pairs of blocks of a set partition. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000732The number of double deficiencies of 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. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St001010Number of indecomposable injective modules with projective dimension g-1 when g is 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. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001513The number of nested exceedences of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001903The number of fixed points of a parking function. St000056The decomposition (or block) number of a permutation. St000060The greater neighbor of the maximum. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000307The number of rowmotion orbits of a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in 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. St001162The minimum jump of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001946The number of descents in a parking function. St000469The distinguishing number of a graph. St000542The number of left-to-right-minima of a permutation. St000767The number of runs in an integer composition. St000823The number of unsplittable factors of the set partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St000903The number of different parts of an integer composition. St000983The length of the longest alternating subword. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001516The number of cyclic bonds of a permutation. St001884The number of borders of a binary word. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001644The dimension of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation.