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Your data matches 551 different statistics following compositions of up to 3 maps.
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Matching statistic: St001258
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
St001258: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 2 = 0 + 2
[1,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> 4 = 2 + 2
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.
For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
Matching statistic: St000155
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => 0
[1,1,0,0]
=> [2,1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,3,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => 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: St001687
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St000619
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,2,1,4] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,3,2,4] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 3 = 2 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000702
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => 2 = 0 + 2
[1,1,0,0]
=> [1,2] => [1,2] => 2 = 0 + 2
[1,0,1,0,1,0]
=> [2,1,3] => [1,3,2] => 2 = 0 + 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 3 = 1 + 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,4,2,3] => 3 = 1 + 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,3,2,4] => 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 2 = 0 + 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,4,2,3] => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4 = 2 + 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [1,4,2,3] => 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [1,2,4,3] => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,3,4,2] => 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 3 = 1 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 4 = 2 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,3,5,2,4] => 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => 4 = 2 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,3,2,4,5] => 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,5,2,3,4] => 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,3,4,2,5] => 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,4,2,3,5] => 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => 3 = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 5 = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [1,4,2,5,3] => 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [1,2,5,3,4] => 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [1,4,5,2,3] => 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [1,5,2,3,4] => 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [1,5,2,4,3] => 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [1,2,4,3,5] => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [1,2,5,3,4] => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,5,2,4] => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [1,2,4,5,3] => 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [1,5,2,3,4] => 4 = 2 + 2
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000725
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => 2 = 0 + 2
[1,1,0,0]
=> [1,2] => [2,1] => 2 = 0 + 2
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => 3 = 1 + 2
[1,0,1,1,0,0]
=> [2,1,3] => [3,2,1] => 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,3,2] => [2,1,3] => 2 = 0 + 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 3 = 1 + 2
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,2,3,1] => 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,2,1,4] => 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,1,3,4] => 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,4,3,1] => 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2,4] => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,4,2,1] => 4 = 2 + 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,3,1,4] => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,4,1,3] => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,2,3,4,1] => 4 = 2 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,2,3,1,5] => 3 = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,2,3,5,1] => 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2,1,4,5] => 3 = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,3,4] => 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2,5,1,4] => 3 = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,1,3,4,5] => 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,5,3,4,1] => 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [2,5,3,1,4] => 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,2,4,5] => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,5,2,4,1] => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,1,2,3,5] => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,5,2,3,1] => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,4,2,1,5] => 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,5,2,1,4] => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,5,2,1,3] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,4,2,5,1] => 4 = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,3,1,4,5] => 3 = 1 + 2
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000991
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 2 = 0 + 2
[1,1,0,0]
=> [2,1] => [1,2] => 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2 = 0 + 2
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => 2 = 0 + 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 3 = 1 + 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 3 = 1 + 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,2,3] => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 3 = 1 + 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,4,2,3] => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,4,2,3] => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 4 = 2 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,3,4] => 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,2,4] => 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,2,5,3] => 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,2,3,5] => 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,5,2,3] => 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,2,3,4] => 4 = 2 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,2,4] => 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,4,5,2,3] => 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,2,3,4] => 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,5,2,3,4] => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,5,2,4,3] => 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,3,5] => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,2,4,5,3] => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 4 = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,4,5,2,3] => 3 = 1 + 2
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000021
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [2,1] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,4,1] => [2,3,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,3,1,4] => [2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,4,2,1] => [3,2,1] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,4,1] => [3,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,1,3,2] => [1,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3,1,4,5] => [2,3,1,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,4,2,5,1] => [3,4,2,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [2,4,1,3] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,5,2,4,1] => [3,2,4,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,4,2,1,5] => [3,4,2,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,5,2,1,4] => [3,2,1,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,1,4,2] => [3,1,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,5,3,2,1] => [4,3,2,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,5,1] => [3,2,4,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,4,1,5] => [3,2,4,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [3,2,1,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,3,5,2,1] => [4,3,2,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,5,1] => [4,2,3,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [2,3,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [4,2,3,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,3,4,2,1] => [3,4,2,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000242
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000242: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000242: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => [2,1] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,3,2,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,5,4,3,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,1,5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,4,3,5,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,5,4,3,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => 2
Description
The number of indices that are not cyclical small weak excedances.
A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.
Matching statistic: St000337
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1] => [1] => 0
[1,1,0,0]
=> [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [2,3,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [2,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,1] => [3,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [2,3,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [1,3,4,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [3,4,1,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [3,4,1,2] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => [2,3,1,4] => 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 $\tau_{i}$.
The following 541 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001388The number of non-attacking neighbors of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001864The number of excedances of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000245The number of ascents of a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000331The number of upper interactions of a Dyck path. St000470The number of runs in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000672The number of minimal elements in Bruhat order not less than the permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000007The number of saliances of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000203The number of external nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000831The number of indices that are either descents or recoils. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000288The number of ones in a binary word. St000292The number of ascents of a binary word. St000390The number of runs of ones in a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001405The number of bonds in a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000834The number of right outer peaks of a permutation. St000291The number of descents of a binary word. St001330The hat guessing number of a graph. St000392The length of the longest run of ones in a binary word. St000454The largest eigenvalue of a graph if it is integral. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St000640The rank of the largest boolean interval in a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000982The length of the longest constant subword. St001863The number of weak excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000144The pyramid weight of the Dyck path. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001488The number of corners of a skew partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St000455The second largest eigenvalue of a graph if it is integral. St000742The number of big ascents of a permutation after prepending zero. St000259The diameter of a connected graph. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000381The largest part of an integer composition. St000013The height of a Dyck path. St001964The interval resolution global dimension of a poset. St001090The number of pop-stack-sorts needed to sort a permutation. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000389The number of runs of ones of odd length in a binary word. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001720The minimal length of a chain of small intervals in a lattice. St000628The balance of a binary word. St000260The radius of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000035The number of left outer peaks of a permutation. St000648The number of 2-excedences of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001638The book thickness of a graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000201The number of leaf nodes in a binary tree. St000451The length of the longest pattern of the form k 1 2. St001520The number of strict 3-descents. St001960The number of descents of a permutation minus one if its first entry is not one. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000682The Grundy value of Welter's game on a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001556The number of inversions of the third entry of a permutation. St001712The number of natural descents of a standard Young tableau. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St000039The number of crossings of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001884The number of borders of a binary word. St001935The number of ascents in a parking function. St000097The order of the largest clique of the graph. St000120The number of left tunnels of a Dyck path. St000904The maximal number of repetitions of an integer composition. St001424The number of distinct squares in a binary word. St001581The achromatic number of a graph. St001668The number of points of the poset minus the width of the poset. 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. St000735The last entry on the main diagonal of a standard tableau. St001180Number of indecomposable injective modules with projective dimension at most 1. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000386The number of factors DDU in a Dyck path. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000632The jump number of the poset. 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. St000298The order dimension or Dushnik-Miller dimension of a poset. St000327The number of cover relations in a poset. St000567The sum of the products of all pairs of parts. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001637The number of (upper) dissectors of a poset. St000442The maximal area to the right of an up step of a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001812The biclique partition number of a graph. St001060The distinguishing index of a graph. St000741The Colin de Verdière graph invariant. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001626The number of maximal proper sublattices of a lattice. St001083The number of boxed occurrences of 132 in a permutation. St000352The Elizalde-Pak rank of a permutation. St000568The hook number of a binary tree. St000647The number of big descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000862The number of parts of the shifted shape of a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001335The cardinality of a minimal cycle-isolating set of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000402Half the size of the symmetry class of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000080The rank of the poset. St000098The chromatic number of a graph. St000522The number of 1-protected nodes of a rooted tree. St001096The size of the overlap set of a permutation. St001623The number of doubly irreducible elements of a lattice. St001733The number of weak left to right maxima of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000528The height of a poset. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000884The number of isolated descents of a permutation. St001394The genus of a permutation. St000153The number of adjacent cycles of a permutation. St000492The rob statistic of a set partition. St000651The maximal size of a rise in a permutation. St000839The largest opener of a set partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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 $eA$. St001199The dominant dimension of $eAe$ for 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 $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000022The number of fixed points of a permutation. St000356The number of occurrences of the pattern 13-2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001487The number of inner corners of a skew partition. St000891The number of distinct diagonal sums of a permutation matrix. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000562The number of internal points of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000770The major index of an integer partition when read from bottom to top. St000782The indicator function of whether a given perfect matching is an L & P matching. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001396Number of triples of incomparable elements in a finite poset. St001541The Gini index of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001866The nesting alignments of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001115The number of even descents of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000031The number of cycles in the cycle decomposition of a permutation. St001893The flag descent of a signed permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001372The length of a longest cyclic run of ones of a binary word. St000028The number of stack-sorts needed to sort a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000374The number of exclusive right-to-left minima of a permutation. St000422The energy of a graph, if it is integral. St000493The los statistic of a set partition. St001050The number of terminal closers of a set partition. St001075The minimal size of a block of a set partition. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001822The number of alignments of a signed permutation. St000023The number of inner peaks of a permutation. St000779The tier of a permutation. St001469The holeyness of a permutation. St000099The number of valleys of a permutation, including the boundary. St001517The length of a longest pair of twins in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000233The number of nestings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. 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. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. 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. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001153The number of blocks with even minimum in a set 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 $K[x]/(x^n)$. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001403The number of vertical separators in a permutation. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001811The Castelnuovo-Mumford regularity of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000221The number of strong fixed points of a permutation. St000254The nesting number of a set partition. St000272The treewidth of a graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys of a permutation. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000536The pathwidth of a graph. St000570The Edelman-Greene number of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000654The first descent of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000899The maximal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001114The number of odd descents of a permutation. 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. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001220The width of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001423The number of distinct cubes in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001569The maximal modular displacement 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. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001768The number of reduced words of a signed permutation. St001781The interlacing number of a set partition. St001792The arboricity of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001823The Stasinski-Voll length of a signed permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001946The number of descents in a parking function. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000092The number of outer peaks of a permutation. St000163The size of the orbit of the set partition under rotation. St000166The depth minus 1 of an ordered tree. St000172The Grundy number of a graph. St000209Maximum difference of elements in cycles. St000236The number of cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000495The number of inversions of distance at most 2 of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000652The maximal difference between successive positions of a permutation. St000730The maximal arc length of a set partition. St000822The Hadwiger number of the graph. St000824The sum of the number of descents and the number of recoils of a permutation. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001516The number of cyclic bonds of a permutation. St001557The number of inversions of the second entry of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001580The acyclic chromatic number of a graph. St001642The Prague dimension of a graph. St001667The maximal size of a pair of weak twins for a permutation. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001716The 1-improper chromatic number of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001769The reflection length of a signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001963The tree-depth of a graph. St000485The length of the longest cycle of a permutation. St000488The number of cycles of a permutation of length at most 2. St000503The maximal difference between two elements in a common block. St000844The size of the largest block in the direct sum decomposition of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000737The last entry on the main diagonal of a semistandard tableau. St000956The maximal displacement of a permutation. St001267The length of the Lyndon factorization of the binary word. St000170The trace of a semistandard tableau. St001621The number of atoms of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001875The number of simple modules with projective dimension at most 1. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size 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. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000527The width of the poset. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001268The size of the largest ordinal summand in the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001717The largest size of an interval in a poset.
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