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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St000052
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000871
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0]
=> [3,1,2] => [2,3,1] => [1,2,3] => 0
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [1,2,4,3] => 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,4,2,1] => [1,3,2,4] => 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => [1,2,4,5,3] => 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [2,4,5,3,1] => [1,2,4,3,5] => 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => [1,2,5,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [1,3,4,5,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [1,3,5,4,2] => 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => [1,3,2,4,5] => 0
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => [1,3,5,2,4] => 0
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,5,2,1,4] => [1,3,2,5,4] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [1,4,5,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,1,5,3,2] => [1,4,3,5,2] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,4,3,2,5] => 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,4,6,1,5] => [1,2,3,4,6,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,3,5,1,6,4] => [1,2,3,5,6,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,3,5,6,4,1] => [1,2,3,5,4,6] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,3,6,1,4,5] => [1,2,3,6,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [2,4,1,5,6,3] => [1,2,4,5,6,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [2,4,1,6,3,5] => [1,2,4,6,5,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [2,4,5,3,6,1] => [1,2,4,3,5,6] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [2,4,5,6,3,1] => [1,2,4,6,3,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,4,6,3,1,5] => [1,2,4,3,6,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [2,5,1,3,6,4] => [1,2,5,6,4,3] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [2,5,1,6,4,3] => [1,2,5,4,6,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [2,5,6,3,4,1] => [1,2,5,4,3,6] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [2,6,1,3,4,5] => [1,2,6,5,4,3] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,1,4,6,2,5] => [1,3,4,6,5,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [3,1,5,2,6,4] => [1,3,5,6,4,2] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [3,1,5,6,4,2] => [1,3,5,4,6,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,1,6,2,4,5] => [1,3,6,5,4,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => [1,3,2,4,5,6] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,4,2,6,1,5] => [1,3,2,4,6,5] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => [1,3,5,6,2,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [3,4,6,2,1,5] => [1,3,6,5,2,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,5,2,1,6,4] => [1,3,2,5,6,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,5,2,6,4,1] => [1,3,2,5,4,6] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,5,6,2,4,1] => [1,3,6,2,5,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,6,2,1,4,5] => [1,3,2,6,5,4] => 1
Description
The number of very big ascents of a permutation.
A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$.
For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
Matching statistic: St000647
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [3,1,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [5,3,2,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [4,2,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [5,4,2,1,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [5,2,1,3,4] => 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,6,4,7] => [1,4,2,6,3,5,7] => [1,6,5,3,2,4,7] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => [3,2,1,4,7,6,5] => ? = 0
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,1,5,3,6,7] => [3,1,5,2,4,6,7] => [5,4,2,1,3,6,7] => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,7,5] => [3,1,5,2,7,4,6] => [7,6,4,2,1,3,5] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,7,4,6] => [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => ? = 3
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,1,2,7,4,5] => [3,4,1,6,7,2,5] => [3,1,6,2,4,7,5] => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,1,7,2,3,4,6] => [2,4,5,6,1,7,3] => [7,3,5,1,2,4,6] => ? = 2
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St000065
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [[1]]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,1,0,1,1,0,0,0]
=> [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
[1,1,1,0,0,0,1,0]
=> [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
[1,1,1,0,0,1,0,0]
=> [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
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[1,1,1,1,0,0,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [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]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [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]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [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]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [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]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [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]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [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]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [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]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [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]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [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]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [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]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [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
[1,1,0,1,0,0,1,0,1,0]
=> [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]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [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]]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [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]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [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]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [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
[1,1,0,1,1,0,0,0,1,0]
=> [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
[1,1,0,1,1,0,0,1,0,0]
=> [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
[1,1,0,1,1,0,1,0,0,0]
=> [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]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [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
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0,0],[1,0,0,0,0,-1,0,1],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,-1,1],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,-1,0,1],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> ? = 1
[]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,-1,1],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,-1,1],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0]]
=> ? = 1
Description
The number of entries equal to -1 in an alternating sign matrix.
The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St001167
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1
[]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St000356
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [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] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,2,1,6,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [3,2,5,4,1,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [3,4,2,6,5,1,7] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => ? = 0
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [3,4,5,2,7,6,1] => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,6,5,7,4,1] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,4,6,3,7,2] => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,5,7,8,6] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ? = 1
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St001323
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001323: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001323: Graphs ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,6,3,5,4,2,1,8] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,7,4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [9,8,7,4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[]
=> []
=> [] => ([],0)
=> ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,8,7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [10,9,8,7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
Description
The independence gap of a graph.
This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph.
In particular, this statistic is $0$ for well covered graphs
Matching statistic: St001083
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [4,5,1,2,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [5,4,1,2,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,5,2,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,4,5,2,3] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [5,1,4,2,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,5,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => [4,5,6,7,1,2,3] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => [5,6,7,4,1,2,3] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [3,2,1,5,6,4,7] => [7,4,6,5,1,2,3] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [6,7,3,5,4,1,2] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 0
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,6,5] => [5,6,7,2,4,3,1] => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,4,6,2,7] => [7,2,6,4,5,3,1] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => [1,7,5,6,3,4,2] => ? = 3
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,4,3,7,1] => [1,7,3,4,6,5,2] => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => [1,7,2,4,5,6,3] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [7,1,2,3,5,6,4] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [7,1,2,3,4,6,5] => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => [1,7,2,3,4,6,5] => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,6,4,3,2,1,8] => [8,1,2,3,4,6,7,5] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,8,7,5,4,3,2,1] => [1,2,3,4,5,7,8,6] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [8,1,2,3,4,5,7,6] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [7,9,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,9,7] => ? = 1
Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St000371
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5,1,4,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5,3,1,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [5,4,1,2,3] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => [3,1,2,7,4,5,6] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => [3,1,2,4,7,5,6] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [3,1,2,5,6,4,7] => [3,1,2,6,5,4,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,6,4,7] => [1,6,3,5,2,4,7] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,7,5] => [7,2,4,1,6,3,5] => ? = 3
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,1,7,3,4,5] => [7,2,6,1,3,4,5] => ? = 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,1,3,7,4] => [7,2,1,6,5,3,4] => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => [7,1,3,2,6,4,5] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => [7,1,2,4,3,5,6] => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => [6,1,2,4,3,5,7] => ? = 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,1,7,2,3,5] => [7,6,1,4,2,3,5] => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => [7,1,2,3,5,4,6] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,1,2,3,7,4] => [7,1,2,6,5,3,4] => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [7,1,2,3,5,4,6,8] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => [8,1,2,3,4,6,5,7] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [7,1,2,3,4,6,5,8] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [8,1,2,3,4,5,7,6] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [7,9,1,2,3,4,5,6,8] => [9,1,2,3,4,5,7,6,8] => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [9,1,2,3,4,5,6,8,7] => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [10,1,2,3,4,5,6,7,9,8] => ? = 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000373
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5,1,4,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5,3,1,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [5,4,1,2,3] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => [3,1,2,7,4,5,6] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => [3,1,2,4,7,5,6] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [3,1,2,5,6,4,7] => [3,1,2,6,5,4,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,6,4,7] => [1,6,3,5,2,4,7] => ? = 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,7,5] => [7,2,4,1,6,3,5] => ? = 3
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,6,1,7,3,4,5] => [7,2,6,1,3,4,5] => ? = 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,1,3,7,4] => [7,2,1,6,5,3,4] => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => [7,1,3,2,6,4,5] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => [7,1,2,4,3,5,6] => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => [6,1,2,4,3,5,7] => ? = 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,1,7,2,3,5] => [7,6,1,4,2,3,5] => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => [7,1,2,3,5,4,6] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,1,2,3,7,4] => [7,1,2,6,5,3,4] => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [7,1,2,3,5,4,6,8] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => [8,1,2,3,4,6,5,7] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [7,1,2,3,4,6,5,8] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [8,1,2,3,4,5,7,6] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [7,9,1,2,3,4,5,6,8] => [9,1,2,3,4,5,7,6,8] => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [9,1,2,3,4,5,6,8,7] => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [10,1,2,3,4,5,6,7,9,8] => ? = 1
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000053The number of valleys of the Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000358The number of occurrences of the pattern 31-2. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000451The length of the longest pattern of the form k 1 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000015The number of peaks of a Dyck path. St000872The number of very big descents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. 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 $eA$. St001095The number of non-isomorphic posets with precisely one further covering relation. St000920The logarithmic height of a 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. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
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