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Your data matches 67 different statistics following compositions of up to 3 maps.
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Matching statistic: St000340
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns 110 and 001.
Matching statistic: St001036
(load all 155 compositions to match this statistic)
(load all 155 compositions to match this statistic)
St001036: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 2 = 3 - 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]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 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]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 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]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[]
=> ? = 0 - 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001499
(load all 65 compositions to match this statistic)
(load all 65 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
St001499: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[]
=> []
=> ? = 0
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000249
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> [1] => {{1}}
=> ? = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 2 = 1 + 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] => {{1,5},{2,4},{3}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 4 = 3 + 1
[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] => {{1,2,5},{3,4}}
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[]
=> []
=> [] => {}
=> ? = 0 + 1
Description
The number of singletons ([[St000247]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000292
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 74%●distinct values known / distinct values provided: 71%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 74%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0]
=> [1]
=> 10 => 00 => 0 = 1 - 1
[1,1,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0000 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 001 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 010 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 00 => 0 = 1 - 1
[1,1,1,0,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 000000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 001010 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01000 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0101 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 00010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 001 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0110 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 010 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 00 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 00000000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0000001 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 00001010 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0000100 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 000011 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 00101000 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0010101 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 00100010 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0010000 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 001001 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 00110110 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0011010 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 001100 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 00111 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0100000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 010001 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0101010 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 010100 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 01011 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0001000 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 000101 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0000010 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 000000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0010110 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 001010 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 011000 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 01101 => 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 010010 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01000 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => 0011011000 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => 0011010010 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 1000101110 => 0011000110 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1000011110 => 0011101110 => ? = 3 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> 101010101010 => 000000000000 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> 10110101010 => ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> 11001101010 => ? => ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> 100101101010 => ? => ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> 1011101010 => ? => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> 100110011010 => ? => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> 10110011010 => ? => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> 101001011010 => ? => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> 11001011010 => ? => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> 100101011010 => ? => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 10101011010 => ? => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> 1011011010 => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> 1100111010 => ? => ? = 4 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> 100010111010 => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2,1]
=> 10010111010 => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> 1010111010 => ? => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> 10110100110 => ? => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> 101001100110 => ? => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> 11001100110 => ? => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> 100101100110 => ? => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> 10101100110 => ? => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1,1]
=> 1011100110 => ? => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> 101010010110 => ? => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> 11010010110 => ? => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> 100110010110 => ? => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> 10110010110 => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> 101001010110 => ? => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> 11001010110 => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> 100101010110 => ? => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => ? => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> 1011010110 => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1,1]
=> 1100110110 => ? => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1,1]
=> 100010110110 => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,1,1]
=> 10010110110 => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> 1010110110 => ? => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> 100011001110 => ? => ? = 5 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1,1]
=> 10011001110 => ? => ? = 5 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> 1011001110 => ? => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,1,1]
=> 101000101110 => ? => ? = 3 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 74%●distinct values known / distinct values provided: 71%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 74%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0]
=> [1]
=> 10 => 00 => 0 = 1 - 1
[1,1,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0000 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 001 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 010 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 00 => 0 = 1 - 1
[1,1,1,0,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 000000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 001010 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01000 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0101 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 00010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 001 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0110 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 010 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 00 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 00000000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0000001 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 00001010 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0000100 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 000011 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 00101000 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0010101 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 00100010 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0010000 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 001001 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 00110110 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0011010 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 001100 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 00111 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0100000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 010001 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0101010 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 010100 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 01011 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0001000 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 000101 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0000010 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 000000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0010110 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 001010 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 011000 => 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 01101 => 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 010010 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01000 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => 0011011000 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => 0011010010 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 1000101110 => 0011000110 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1000011110 => 0011101110 => ? = 3 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> 101010101010 => 000000000000 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> 10110101010 => ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> 11001101010 => ? => ? = 4 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> 100101101010 => ? => ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => ? => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> 1011101010 => ? => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> 100110011010 => ? => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> 10110011010 => ? => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> 101001011010 => ? => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> 11001011010 => ? => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> 100101011010 => ? => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 10101011010 => ? => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> 1011011010 => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> 1100111010 => ? => ? = 4 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> 100010111010 => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2,1]
=> 10010111010 => ? => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> 1010111010 => ? => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> 10110100110 => ? => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> 101001100110 => ? => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> 11001100110 => ? => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> 100101100110 => ? => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> 10101100110 => ? => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1,1]
=> 1011100110 => ? => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> 101010010110 => ? => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> 11010010110 => ? => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> 100110010110 => ? => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> 10110010110 => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> 101001010110 => ? => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> 11001010110 => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> 100101010110 => ? => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 10101010110 => ? => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> 1011010110 => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1,1]
=> 1100110110 => ? => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1,1]
=> 100010110110 => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,1,1]
=> 10010110110 => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> 1010110110 => ? => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> 100011001110 => ? => ? = 5 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1,1]
=> 10011001110 => ? => ? = 5 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,3,1,1,1]
=> 1011001110 => ? => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,1,1]
=> 101000101110 => ? => ? = 3 - 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000619
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 71%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [1,2] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => 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] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => 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] => 1
[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] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => 3
[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] => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [1,2,5,3,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => 3
[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] => [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [1,5,2,4,3] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,5,4,2,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,5,2,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,5,3,7,1] => [1,7,3,5,6,4,2] => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => [1,7,6,4,5,3,2] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,6,4,7,1] => [1,7,4,6,5,3,2] => ? = 5
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => [1,7,4,5,6,3,2] => ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [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] => ? = 4
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,5,1] => [1,5,7,6,4,3,2] => ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,6,4,1] => [1,4,6,7,5,3,2] => ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => [1,5,6,7,3,4,2] => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => [1,7,6,2,5,4,3] => ? = 5
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,1] => [1,7,6,5,2,4,3] => ? = 5
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,4,2,6,5,7,1] => [1,7,5,6,2,4,3] => ? = 4
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,7,6,1] => [1,6,7,4,5,2,3] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => [1,6,7,5,4,2,3] => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,2,4,6,7,5,1] => [1,5,7,6,4,2,3] => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 3
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,4,2,7,6,5,1] => [1,5,6,7,2,4,3] => ? = 3
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => [1,7,6,2,3,5,4] => ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,3,5,2,7,6,1] => [1,6,7,2,5,3,4] => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,3,2,5,7,6,1] => [1,6,7,5,2,3,4] => ? = 3
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,3,2,6,7,5,1] => [1,5,7,6,2,3,4] => ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => [1,5,6,7,2,3,4] => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => [1,6,7,2,3,4,5] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,4,6,5,3,2,7] => [7,2,3,5,6,4,1] => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [7,2,6,5,4,3,1] => ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,5,6,4,2,7] => [7,2,4,6,5,3,1] => ? = 4
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => [7,2,4,5,6,3,1] => ? = 3
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,3,6,5,2,7] => [7,2,5,6,3,4,1] => ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [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] => ? = 4
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => ? = 5
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => ? = 4
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => ? = 3
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => ? = 4
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => [7,6,5,2,4,3,1] => ? = 5
[1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 4
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [7,4,6,5,2,3,1] => ? = 4
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [7,4,5,6,2,3,1] => ? = 3
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,3,2,6,5,7] => [7,5,6,2,3,4,1] => ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [2,7,3,4,5,6,1] => ? = 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,3,7,2] => [2,7,3,6,5,4,1] => ? = 4
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,6,5,3,7,2] => [2,7,3,5,6,4,1] => ? = 3
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,4,6,7,2] => [2,7,6,4,5,3,1] => ? = 4
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => ? = 5
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,4,7,2] => [2,7,4,6,5,3,1] => ? = 4
Description
The number of cyclic descents of a permutation.
For a permutation π of {1,…,n}, this is given by the number of indices 1≤i≤n such that π(i)>π(i+1) where we set π(n+1)=π(1).
Matching statistic: St000291
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 71%
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> 10 => 10 => 1
[1,0,1,0]
=> 1010 => 1100 => 1
[1,1,0,0]
=> 1100 => 0110 => 1
[1,0,1,0,1,0]
=> 101010 => 111000 => 1
[1,0,1,1,0,0]
=> 101100 => 011010 => 2
[1,1,0,0,1,0]
=> 110010 => 101100 => 2
[1,1,0,1,0,0]
=> 110100 => 011100 => 1
[1,1,1,0,0,0]
=> 111000 => 001110 => 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11110000 => 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 01110010 => 2
[1,0,1,1,0,0,1,0]
=> 10110010 => 10110100 => 3
[1,0,1,1,0,1,0,0]
=> 10110100 => 01110100 => 2
[1,0,1,1,1,0,0,0]
=> 10111000 => 00110110 => 2
[1,1,0,0,1,0,1,0]
=> 11001010 => 11011000 => 2
[1,1,0,0,1,1,0,0]
=> 11001100 => 01011010 => 3
[1,1,0,1,0,0,1,0]
=> 11010010 => 10111000 => 2
[1,1,0,1,0,1,0,0]
=> 11010100 => 01111000 => 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 00111010 => 2
[1,1,1,0,0,0,1,0]
=> 11100010 => 10011100 => 2
[1,1,1,0,0,1,0,0]
=> 11100100 => 01011100 => 2
[1,1,1,0,1,0,0,0]
=> 11101000 => 00111100 => 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 00011110 => 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1111100000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0111100010 => 2
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1011100100 => 3
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0111100100 => 2
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0011100110 => 2
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101101000 => 3
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0101101010 => 4
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 1011101000 => 3
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0111101000 => 2
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011101010 => 3
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1001101100 => 3
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0101101100 => 3
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101100 => 2
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0001101110 => 2
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1110110000 => 2
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0110110010 => 3
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1010110100 => 4
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0110110100 => 3
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0010110110 => 3
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1101110000 => 2
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0101110010 => 3
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1011110000 => 2
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0111110000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 0011110010 => 2
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1001110100 => 3
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0101110100 => 3
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 0011110100 => 2
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0001110110 => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => 11111110000000 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => 01111110000100 => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => 01011110001010 => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => 01111110001000 => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => 00111110001100 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 10101100110010 => 10101110010100 => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> 10101100110100 => 01101110010100 => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> 10101101001010 => 11011110010000 => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 10101101001100 => 01011110010010 => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 10101101010100 => 01111110010000 => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> 10101101101000 => 00111110010100 => ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 10101110011000 => 00101110011010 => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 10101110100010 => 10011110011000 => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> 10101110100100 => 01011110011000 => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10101110101000 => 00111110011000 => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => 00011110011100 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> 10110010110100 => 01110110100100 => ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 10110011001010 => 11010110101000 => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 10110011001100 => 01010110101010 => ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> 10110011010010 => 10110110101000 => ? = 5
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 10110011010100 => 01110110101000 => ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> 10110011101000 => 00110110101100 => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> 10110100101010 => 11101110100000 => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> 10110100101100 => 01101110100010 => ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> 10110100110010 => 10101110100100 => ? = 5
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> 10110100110100 => 01101110100100 => ? = 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> 10110101001010 => 11011110100000 => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> 10110101001100 => 01011110100010 => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => 01111110100000 => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> 10110101101000 => 00111110100100 => ? = 3
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> 10110110011000 => 00101110101010 => ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> 10110110100010 => 10011110101000 => ? = 4
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> 10110110100100 => 01011110101000 => ? = 4
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> 10110110101000 => 00111110101000 => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> 10110111010000 => 00011110101100 => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> 10111001100010 => 10010110110100 => ? = 5
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> 10111001100100 => 01010110110100 => ? = 5
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> 10111001101000 => 00110110110100 => ? = 4
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> 10111010001010 => 11001110110000 => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> 10111010001100 => 01001110110010 => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> 10111010010010 => 10101110110000 => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> 10111010010100 => 01101110110000 => ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> 10111010011000 => 00101110110010 => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> 10111010100010 => 10011110110000 => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> 10111010100100 => 01011110110000 => ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> 10111010101000 => 00111110110000 => ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 10111011010000 => 00011110110100 => ? = 3
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> 10111100110000 => 00010110111010 => ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> 10111101000010 => 10001110111000 => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> 10111101000100 => 01001110111000 => ? = 3
Description
The number of descents of a binary word.
Matching statistic: St000925
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 71%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2 = 1 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> {{1},{2,4,6},{3},{5}}
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1},{2,4,5,6},{3}}
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> {{1},{2,5,6},{3,4}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3,6},{4,5}}
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4,6},{5}}
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3,4,6},{5}}
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5,6}}
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> {{1},{2,3},{4,6},{5}}
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6}}
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> {{1,3},{2},{4,5},{6}}
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> {{1,3,4},{2},{5},{6}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> {{1,2},{3,6},{4,5}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> {{1,2},{3},{4,6},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> {{1,2},{3,4,6},{5}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> {{1,3},{2},{4},{5,6}}
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> {{1,3},{2},{4,6},{5}}
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> {{1,3},{2},{4,5,6}}
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> {{1,3,4},{2},{5,6}}
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,3,6,8},{4},{5},{7}}
=> ? = 4 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,5,6,8},{4},{7}}
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> {{1},{2,3,6,7,8},{4,5}}
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> {{1},{2,5,8},{3},{4},{6},{7}}
=> ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> {{1},{2,5,8},{3},{4},{6,7}}
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> {{1},{2,4,5,8},{3},{6,7}}
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,4,5,6,8},{3},{7}}
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2,5,6,7,8},{3},{4}}
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> {{1},{2,6,8},{3,5},{4},{7}}
=> ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> {{1},{2,5,7,8},{3,4},{6}}
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> {{1},{2,5,6,8},{3,4},{7}}
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2,5,6,7,8},{3,4}}
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> {{1},{2,6,7,8},{3,4,5}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> {{1},{2},{3,8},{4},{5,6,7}}
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6},{7}}
=> ? = 5 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 6 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,7},{6}}
=> ? = 5 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> {{1},{2},{3},{4,8},{5,6,7}}
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3,4},{5,8},{6,7}}
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> {{1},{2},{3,4,8},{5,6,7}}
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> {{1},{2},{3,4,8},{5},{6,7}}
=> ? = 4 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> {{1},{2},{3,5,8},{4},{6},{7}}
=> ? = 5 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> {{1},{2},{3,5,8},{4},{6,7}}
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> {{1},{2},{3,4,5,8},{6,7}}
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2},{3,4,6,8},{5},{7}}
=> ? = 4 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,8},{7}}
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3,5,6,7,8},{4}}
=> ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,8},{7}}
=> ? = 5 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2},{3},{4,5,7,8},{6}}
=> ? = 4 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2},{3},{4,5,6,8},{7}}
=> ? = 4 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4},{5,6,7,8}}
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3},{5},{6,8},{7}}
=> ? = 5 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> {{1},{2,4},{3},{5,8},{6},{7}}
=> ? = 5 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> {{1},{2,4},{3},{5,8},{6,7}}
=> ? = 4 + 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> {{1},{2,3},{4,7,8},{5,6}}
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,3},{4},{5,7,8},{6}}
=> ? = 4 + 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> {{1},{2,3},{4,5,8},{6},{7}}
=> ? = 4 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3},{4,5,8},{6,7}}
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3},{4},{5,6,8},{7}}
=> ? = 4 + 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3},{4,5,7,8},{6}}
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3},{4,5,6,8},{7}}
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 2 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,4},{3},{5,6,7,8}}
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> {{1},{2,3,5},{4},{6,8},{7}}
=> ? = 4 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2,3,4},{5},{6,7,8}}
=> ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> {{1},{2,3,4},{5,7,8},{6}}
=> ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,4},{5,6,8},{7}}
=> ? = 3 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 2 + 1
Description
The number of topologically connected components of a set partition.
For example, the set partition {{1,5},{2,3},{4,6}} has the two connected components {1,4,5,6} and {2,3}.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000053
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 86%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 5
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 5
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 5
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 4
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 5
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 3
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
Description
The number of valleys of the Dyck path.
The following 57 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000105The number of blocks in the set partition. St000636The hull number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000871The number of very big ascents of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000167The number of leaves of an ordered tree. St001083The number of boxed occurrences of 132 in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000159The number of distinct parts of the integer partition. St000824The sum of the number of descents and the number of recoils of a permutation. St001488The number of corners of a skew partition. St000164The number of short pairs. St000157The number of descents of a standard tableau. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000052The number of valleys of a Dyck path not on the x-axis. 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. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001960The number of descents of a permutation minus one if its first entry is not one. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000035The number of left outer peaks of a permutation. St000317The cycle descent number of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001637The number of (upper) dissectors of a poset.
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