- FindStat

Your data matches 198 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000711
(load all 3 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of big exceedences of a permutation. A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$. This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.

Matching statistic: St000155
(load all 5 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].

Matching statistic: St000375

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].

Matching statistic: St000710
(load all 5 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of big deficiencies of a permutation. A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$. This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.

Matching statistic: St000996
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000337
(load all 2 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.

Matching statistic: St000703
(load all 6 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].

Matching statistic: St000052
(load all 2 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.

Matching statistic: St000053
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St000374
(load all 4 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].

The following 188 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000562The number of internal points of a set partition. St000647The number of big descents of a permutation. St000872The number of very big descents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000451The length of the longest pattern of the form k 1 2. St000619The number of cyclic descents of a permutation. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001638The book thickness of a graph. St001427The number of descents of a signed permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000741The Colin de Verdière graph invariant. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000934The 2-degree of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000932The number of occurrences of the pattern UDU in 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000478Another weight of a partition according to Alladi. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000284The Plancherel distribution on integer partitions. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001651The Frankl number of a lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000225Difference between largest and smallest parts in a partition. St000031The number of cycles in the cycle decomposition of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St001520The number of strict 3-descents. St001864The number of excedances of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000936The number of even values of the symmetric group character corresponding to the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St001877Number of indecomposable injective modules with projective dimension 2. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001862The number of crossings of a signed permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. 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 $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000068The number of minimal elements in a poset. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001866The nesting alignments of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001490The number of connected components of a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001867The number of alignments of type EN of a signed permutation. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. 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 $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001556The number of inversions of the third entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001768The number of reduced words of a signed permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000679The pruning number of an ordered tree. St001624The breadth of a lattice. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. 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. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000449The number of pairs of vertices of a graph with distance 4. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph. St001271The competition number of a graph.