- FindStat

Your data matches 426 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001264
(load all 65 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001264: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.

Matching statistic: St000234
(load all 4 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of global ascents of a permutation. The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$ Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$ For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$ or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].

Matching statistic: St000317
(load all 4 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cycle descent number of a permutation. Let

$(i_1,\ldots,i_k)$ be a cycle of a permutation

$\pi$ such that

$i_1$ is its smallest element. A **cycle descent** of

$(i_1,\ldots,i_k)$ is an

$i_a$ for

$1 \leq a < k$ such that

$i_a > i_{a+1}$ . The **cycle descent set** of

$\pi$ is then the set of descents in all the cycles of

$\pi$ , and the **cycle descent number** is its cardinality.

Matching statistic: St000355
(load all 5 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$21\!\!-\!\!3$ .

Matching statistic: St000731
(load all 6 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000731: 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]
 => [2,1] => [2,1] => [2,1] => 0
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [2,3,1] => [3,2,1] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => [2,3,1] => 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [2,3,4,1] => [4,2,3,1] => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [2,4,3,1] => [3,2,4,1] => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => [3,1,4,2] => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [3,1,4,5,2] => [5,1,3,4,2] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,5,4,2] => [4,1,3,5,2] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,1,5,3,2] => [5,1,4,2,3] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,1,4,3,2] => [4,1,5,2,3] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,3,4,2] => [3,1,4,5,2] => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [4,1,2,5,3] => [5,1,2,4,3] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [5,1,2,4,3] => [4,1,2,5,3] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,5,2,4] => [5,1,3,2,4] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,1,4,2,5] => [4,1,3,2,5] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,1,3,2,5] => [3,1,4,2,5] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[]
 => [] => [] => [] => 0

Description

The number of double exceedences of a permutation. A double exceedence is an index

$\sigma(i)$ such that

$i < \sigma(i) < \sigma(\sigma(i))$ .

Matching statistic: St000766
(load all 17 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000766: Integer compositions ⟶ 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] => 0 => [2] => 0
[1,1,0,0]
 => [2,1] => 1 => [1,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => 00 => [3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => 01 => [2,1] => 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => [1,2] => 0
[1,1,0,1,0,0]
 => [2,3,1] => 10 => [1,2] => 0
[1,1,1,0,0,0]
 => [3,2,1] => 11 => [1,1,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => [4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => [3,1] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => [2,2] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 010 => [2,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 011 => [2,1,1] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => [1,3] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => [1,2,1] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 100 => [1,3] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 100 => [1,3] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 101 => [1,2,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 110 => [1,1,2] => 0
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 110 => [1,1,2] => 0
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 110 => [1,1,2] => 0
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 111 => [1,1,1,1] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0111 => [2,1,1,1] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => [1,4] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => [1,3,1] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => [1,2,2] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1010 => [1,2,2] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1011 => [1,2,1,1] => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1100 => [1,1,3] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1101 => [1,1,2,1] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1100 => [1,1,3] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1100 => [1,1,3] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 1101 => [1,1,2,1] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1110 => [1,1,1,2] => 0
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1110 => [1,1,1,2] => 0
[1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 1110 => [1,1,1,2] => 0
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1111 => [1,1,1,1,1] => 0
[]
 => [] =>  => [1] => 0

Description

The number of inversions of an integer composition. This is the number of pairs

$(i,j)$ such that

$i < j$ and

$c_i > c_j$ .

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

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001292: 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]
 => [1,0,1,0]
 => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [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]
 => [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]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 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]
 => [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]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 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]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[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]
 => [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]
 => [1,0,1,0,1,0,1,0,1,0]
 => 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]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0

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: St001549
(load all 4 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [2,3,1] => [1,2,3] => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => 0
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => 0
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => 0
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [1,2,6,4,3,5] => 0
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [1,2,5,6,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [1,6,4,2,3,5] => 0
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [1,6,3,4,2,5] => 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [1,5,6,2,3,4] => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [1,2,3,6,4,5] => 0
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => [1,2,6,3,4,5] => 0
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [1,6,2,3,4,5] => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [1,2,6,7,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => [1,2,7,5,6,3,4] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [1,2,7,4,3,5,6] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,5,4,1,6,7,3] => [1,2,5,6,7,3,4] => 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [1,2,3,7,5,4,6] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [2,3,6,5,1,7,4] => [1,2,3,6,7,4,5] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [1,2,7,5,3,4,6] => 0
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [1,2,7,4,5,3,6] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,6,4,5,1,7,3] => [1,2,6,7,3,4,5] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,4,7,6,1,5] => [1,2,3,4,7,5,6] => 0
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,7,5,6,1,4] => [1,2,3,7,4,5,6] => 0
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,7,4,5,6,1,3] => [1,2,7,3,4,5,6] => 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[]
 => [1,0]
 => [2,1] => [1,2] => 0

Description

The number of restricted non-inversions between exceedances. This is for a permutation

$\sigma$ of length

$n$ given by

$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$

Matching statistic: St001810
(load all 36 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001810: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation smaller than its largest moved point.

Matching statistic: St000056
(load all 4 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The decomposition (or block) number of a permutation. For

$\pi \in \mathcal{S}_n$ , this is given by

$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].

The following 416 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000617The number of global maxima of a Dyck path. St000883The number of longest increasing subsequences of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000516The number of stretching pairs of a permutation. St000732The number of double deficiencies of a permutation. St000769The major index of a composition regarded as a word. St001172The number of 1-rises at odd height of a Dyck path. 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. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001552The number of inversions between excedances and fixed points of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001781The interlacing number of a set partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000678The number of up steps after the last double rise of a Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000297The number of leading ones in a binary word. St000733The row containing the largest entry of a standard tableau. 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. St000382The first part of an integer composition. St000383The last part of an integer composition. St000654The first descent of a permutation. St000765The number of weak records in an integer composition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000392The length of the longest run of ones in a binary word. St001933The largest multiplicity of a part in an integer partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000993The multiplicity of the largest part of an integer partition. St000061The number of nodes on the left branch of a binary tree. St001846The number of elements which do not have a complement in the lattice. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St000366The number of double descents of a permutation. St001868The number of alignments of type NE of a signed permutation. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St000563The number of overlapping pairs of blocks of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001875The number of simple modules with projective dimension at most 1. St000068The number of minimal elements in a poset. St000456The monochromatic index of a connected graph. St000022The number of fixed points of a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St000665The number of rafts of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. 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. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001520The number of strict 3-descents. St001811The Castelnuovo-Mumford regularity of a permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000805The number of peaks of the associated bargraph. St001128The exponens consonantiae of a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001267The length of the Lyndon factorization of the binary word. St000091The descent variation of a composition. St000709The number of occurrences of 14-2-3 or 14-3-2. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000217The number of occurrences of the pattern 312 in a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000407The number of occurrences of the pattern 2143 in a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000562The number of internal points of a set partition. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000779The tier of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000873The aix statistic of a permutation. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. 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. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . 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. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000570The Edelman-Greene number of a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000694The number of affine bounded permutations that project to a given permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001490The number of connected components of a skew partition. St001518The number of graphs with the same ordinary spectrum as the given graph. St001568The smallest positive integer that does not appear twice in the partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001949The rigidity index of a graph. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000906The length of the shortest maximal chain in a poset. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St001352The number of internal nodes in the modular decomposition of a graph. St001545The second Elser number of a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite 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. St000879The number of long braid edges in the graph of braid moves of a permutation. St001964The interval resolution global dimension of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000914The sum of the values of the Möbius function of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001330The hat guessing number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000894The trace of an alternating sign matrix. St001260The permanent of an alternating sign matrix. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000039The number of crossings of a permutation. St000089The absolute variation of a composition. St000090The variation of a composition. St000133The "bounce" of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000233The number of nestings of a set partition. St000247The number of singleton blocks of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St000357The number of occurrences of the pattern 12-3. 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. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 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. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000663The number of right floats of a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000750The number of occurrences of the pattern 4213 in 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. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001524The degree of symmetry of a binary word. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001871The number of triconnected components of a graph. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St000021The number of descents of a permutation. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000260The radius of a connected graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000374The number of exclusive right-to-left minima of a permutation. St000527The width of the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000729The minimal arc length of a set partition. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000909The number of maximal chains of maximal size in a poset. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000991The number of right-to-left minima of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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

$K[x]/(x^n)$ . St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001665The number of pure excedances of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001820The size of the image of the pop stack sorting operator. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000401The size of the symmetry class of a permutation. St000417The size of the automorphism group of the ordered tree. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001060The distinguishing index of a graph. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000315The number of isolated vertices of a graph. St000632The jump number of the poset. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001429The number of negative entries in a signed permutation. St001557The number of inversions of the second entry of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001651The Frankl number of a lattice. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001856The number of edges in the reduced word graph of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000096The number of spanning trees of a graph. St000287The number of connected components of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000309The number of vertices with even degree. St000450The number of edges minus the number of vertices plus 2 of a graph. St000454The largest eigenvalue of a graph if it is integral. St000739The first entry in the last row of a semistandard tableau. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000958The number of Bruhat factorizations of 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. St001410The minimal entry of a semistandard tableau. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001828The Euler characteristic of a graph. St001946The number of descents in a parking function. St000134The size of the orbit of an alternating sign matrix under gyration. St000259The diameter of a connected graph. St000822The Hadwiger number of the graph. St000893The number of distinct diagonal sums of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001734The lettericity of a graph. St001893The flag descent of a signed permutation. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001738The minimal order of a graph which is not an induced subgraph of the given graph.