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Your data matches 266 different statistics following compositions of up to 3 maps.
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Matching statistic: St001204
(load all 133 compositions to match this statistic)
(load all 133 compositions to match this statistic)
St001204: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
Description
Call 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.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra.
The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.
Matching statistic: St001217
(load all 91 compositions to match this statistic)
(load all 91 compositions to match this statistic)
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St000745
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000990
(load all 93 compositions to match this statistic)
(load all 93 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => 1
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2
Description
The first ascent of a permutation.
For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$.
For the first descent, see [[St000654]].
Matching statistic: St001185
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001185: Dyck paths ⟶ 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 - 1
[1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Matching statistic: St000007
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000007: 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]
=> [2,1] => 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [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]
=> [2,3,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [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]
=> [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]
=> [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [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]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [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]
=> [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]
=> [1,3,4,5,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [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]
=> [1,2,4,5,3] => 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000326
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 1010 => 0101 => 2
[1,1,0,0]
=> 1100 => 1001 => 1
[1,0,1,0,1,0]
=> 101010 => 010101 => 2
[1,0,1,1,0,0]
=> 101100 => 011001 => 2
[1,1,0,0,1,0]
=> 110010 => 100101 => 1
[1,1,0,1,0,0]
=> 110100 => 101001 => 1
[1,1,1,0,0,0]
=> 111000 => 110001 => 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 2
[1,0,1,0,1,1,0,0]
=> 10101100 => 01011001 => 2
[1,0,1,1,0,0,1,0]
=> 10110010 => 01100101 => 2
[1,0,1,1,0,1,0,0]
=> 10110100 => 01101001 => 2
[1,0,1,1,1,0,0,0]
=> 10111000 => 01110001 => 2
[1,1,0,0,1,0,1,0]
=> 11001010 => 10010101 => 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10100101 => 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10101001 => 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10110001 => 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 11000101 => 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 11001001 => 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 11010001 => 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 11100001 => 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 2
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101011001 => 2
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101100101 => 2
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0101101001 => 2
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101110001 => 2
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0110010101 => 2
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0110011001 => 2
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0110100101 => 2
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0110101001 => 2
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0110110001 => 2
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0111000101 => 2
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0111001001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0111010001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0111100001 => 2
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1001010101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1001011001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1001100101 => 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1001101001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1001110001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1010010101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1010011001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1010100101 => 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1010101001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1010110001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1011000101 => 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1011001001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1011010001 => 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1011100001 => 1
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1100010101 => 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000542
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => 2
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000654
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000654: 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
[1,1,0,0]
=> [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [2,1,3] => [3,1,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [2,3,1] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2,5,1,4] => 1
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St001184
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
The following 256 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000237The number of small exceedances. St000297The number of leading ones in a binary word. St000352The Elizalde-Pak rank of a permutation. St000390The number of runs of ones in a binary word. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000989The number of final rises of a permutation. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000504The cardinality of the first block of a set partition. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000883The number of longest increasing subsequences of a permutation. St000917The open packing number of a graph. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001316The domatic number of a graph. St001481The minimal height of a peak of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000051The size of the left subtree of a binary tree. St000096The number of spanning trees of a graph. St000133The "bounce" of a permutation. St000234The number of global ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000475The number of parts equal to 1 in a partition. St000481The number of upper covers of a partition in dominance order. St000546The number of global descents of a permutation. St000665The number of rafts of a permutation. St000734The last entry in the first row of a standard tableau. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000948The chromatic discriminant of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001479The number of bridges of a graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001691The number of kings in a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001826The maximal number of leaves on a vertex of a graph. St000552The number of cut vertices of a graph. St001948The number of augmented double ascents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001128The exponens consonantiae of a partition. St001432The order dimension of the partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000461The rix statistic of a permutation. St000153The number of adjacent cycles of a permutation. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St000053The number of valleys of the Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000627The exponent of a binary word. St000675The number of centered multitunnels of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001066The number of simple reflexive modules in the corresponding Nakayama 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. St001197The global dimension of $eAe$ for 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 $eA$. 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 $eA$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001420Half the length of a longest factor which is its own reverse-complement of 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. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001597The Frobenius rank of a skew partition. St001732The number of peaks visible from the left. St001884The number of borders of a binary word. St000259The diameter of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001933The largest multiplicity of a part in an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000534The number of 2-rises of a permutation. St000668The least common multiple of the parts of the partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000663The number of right floats of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions 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. St001890The maximum magnitude of the Möbius function of a poset. St001552The number of inversions between excedances and fixed points of a permutation. St001728The number of invisible descents of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001372The length of a longest cyclic run of ones of a binary word. St000366The number of double descents of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001399The distinguishing number of a poset. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000456The monochromatic index of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000221The number of strong fixed points of a permutation. St000338The number of pixed points of a permutation. St000873The aix statistic of a permutation. St000296The length of the symmetric border of a binary word. St000648The number of 2-excedences of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St001820The size of the image of the pop stack sorting operator. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. 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. 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. St000934The 2-degree of an integer partition. 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. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. 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. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. 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. St000894The trace of an alternating sign matrix. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001846The number of elements which do not have a complement in the lattice. St000741The Colin de Verdière graph invariant. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001877Number of indecomposable injective modules with projective dimension 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000214The number of adjacencies of a permutation. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000649The number of 3-excedences of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001050The number of terminal closers of a set partition. St001621The number of atoms of a lattice.
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