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Your data matches 133 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St001241
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St001241: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> 3 = 4 - 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]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 4 = 5 - 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]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 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]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 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]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
Description
The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one.
Matching statistic: St000153
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = 2 - 1
[1,0,1,0]
=> [1,2] => 2 = 3 - 1
[1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,3,2] => 2 = 3 - 1
[1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,1,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0 = 1 - 1
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000678
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St001050
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 4
[1,1,1,1,0,0,0,0]
=> [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}}
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> {{1,4,5},{2,3},{6}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> {{1},{2},{3,4},{5,6}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> {{1,2},{3,6},{4,5}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> {{1},{2,5},{3,4},{6}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> {{1,3,4},{2},{5,6}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5,6}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> {{1,5},{2},{3,4},{6}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> {{1,4},{2,3},{5},{6}}
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> {{1,2,4},{3},{5,6}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> {{1},{2,3},{4,6},{5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> {{1,5},{2,4},{3},{6}}
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> {{1,2},{3},{4,6},{5}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> {{1},{2},{3,5},{4},{6}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> {{1,2,3},{4,6},{5}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> {{1},{2,4},{3},{5,6}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> {{1,4},{2,3},{5,6}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2,3},{4,5},{6}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> {{1,5},{2,3},{4},{6}}
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> {{1,3},{2},{4},{5,6}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6}}
=> 3
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000460
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 2
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {2,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {2,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {2,3,3,4}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {2,3,3,4}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {2,3,3,4}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {2,3,3,4}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {1,2,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000293
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [[1],[]]
=> []
=> => ? = 2
[1,0,1,0]
=> [[1,1],[]]
=> []
=> => ? ∊ {2,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> => ? ∊ {2,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> => ? ∊ {2,3,3,4}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> => ? ∊ {2,3,3,4}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> => ? ∊ {2,3,3,4}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> => ? ∊ {2,3,3,4}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> => ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> => ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> => ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> => ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> => ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> => ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> => ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> => ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1100 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1000 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1010 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 100 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 100 => 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> => ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1010 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1000 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1010 => 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,5,5,5,5,5,5,6,6,6,7}
Description
The number of inversions of a binary word.
Matching statistic: St000395
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 2
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {2,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {2,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,6,6,6,7}
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001018
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 2
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {2,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {2,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,5,5,5,5,5,5,5,6,7}
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001088
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001088: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001088: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 2
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {2,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {2,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,4}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7}
Description
Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
The following 123 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001118The acyclic chromatic index of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000654The first descent of a permutation. St000939The number of characters of the symmetric group whose value on the partition is positive. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001571The Cartan determinant of the integer partition. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St000675The number of centered multitunnels of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000883The number of longest increasing subsequences of a permutation. St000990The first ascent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001782The order of rowmotion on the set of order ideals of a poset. St000528The height of a poset. St000911The number of maximal antichains of maximal size in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. 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 $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000474Dyson's crank of a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001175The size of a partition minus the hook length of the base cell. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001568The smallest positive integer that does not appear twice in the partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000618The number of self-evacuating tableaux of given shape. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001279The sum of the parts of an integer partition that are at least two. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001541The Gini index of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000454The largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000264The girth of a graph, which is not a tree. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000239The number of small weak excedances. St000989The number of final rises of a permutation. St000214The number of adjacencies of a permutation. St000884The number of isolated descents of a permutation.
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