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Your data matches 215 different statistics following compositions of up to 3 maps.
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Matching statistic: St001498
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000352
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 90%●distinct values known / distinct values provided: 75%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 90%●distinct values known / distinct values provided: 75%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4] => 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4] => 0
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,5,6,3,4] => 0
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,6,2,3,4] => 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,6,1,4,5] => 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => 0
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5,7] => ? = 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,7,1,4,5,6] => ? = 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 2
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,5,6,1,2,7,3] => ? = 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,5,6,1,3,7,4] => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,1,4,6,7] => ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,5,6,1,2,7,4] => ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [3,4,6,1,2,7,5] => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => ? = 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St001195
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 70%●distinct values known / distinct values provided: 25%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 70%●distinct values known / distinct values provided: 25%
Values
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,3,2,4,5,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [6,3,2,5,4,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7,3,6,5,4,2] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,7,5,4,3,6,2] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,7,5,4,6,3,2] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,2,5,4,3,1,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [7,2,5,4,3,6,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [7,2,5,4,6,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [7,2,6,5,4,3,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [7,3,2,6,5,4,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,3,4,6,5,2,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,3,5,4,2,7,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [7,3,5,4,2,6,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [7,3,5,4,6,2,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 60%●distinct values known / distinct values provided: 25%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 60%●distinct values known / distinct values provided: 25%
Values
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [.,[[.,[[[.,.],.],[.,.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [.,[[[[[.,.],.],[.,.]],.],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> 0
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],[.,[.,.]]],.]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> 0
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],[.,.]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[[[.,.],[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[[[.,.],[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],[.,.]]],.]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[.,.]],.],.]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[[.,.],[[[.,.],.],[.,.]]],.]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[[.,[[.,.],[.,.]]],[.,.]],.]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[[[.,.],.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[.,.]],.]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[.,.],[.,[.,.]]],[.,.]],.]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [[[[.,.],[[.,.],.]],[.,.]],.]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],[.,.]],.]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],[.,.]],.]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],.]],.]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],[.,.]],.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [[.,.],[[.,[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
=> ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [[.,.],[[[.,[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [[.,.],[[[.,.],[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [[.,[.,.]],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [[.,[[.,.],.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,.]]]],[.,.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001638
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001638: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001638: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,2,4,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,6,4,5,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,2,3,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,5,3,4,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,3,2,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,3,2,5,6,1,7] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,3,2,7,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [7,4,3,2,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,4,6,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => ([(1,2),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,6,4,5,7,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,6,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => ([(1,2),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,7,5,6,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,7,4,5,6,3] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [5,3,2,4,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [5,3,2,4,6,1,7] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [7,3,2,5,4,6,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [5,3,2,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [7,4,3,5,2,6,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [3,2,4,6,5,1,7] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,7,4,6,5,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,2,7,5,6,4,1] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,2,7,4,5,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [7,3,2,4,6,5,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1,7] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [6,3,2,4,5,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [7,4,3,5,6,2,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [7,3,2,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,6,4,3,5,7,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,7,4,3,5,6,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,7,3,5,4,6,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,6,3,4,5,7,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,6,4,3,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 1
Description
The book thickness of a graph.
The book thickness (or pagenumber, or stacknumber) of a graph is the minimal number of pages required for a book embedding of a graph.
Matching statistic: St001603
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 53%●distinct values known / distinct values provided: 25%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 53%●distinct values known / distinct values provided: 25%
Values
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [4,1]
=> [1]
=> ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5]
=> []
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [5,1]
=> [1]
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [4,2]
=> [2]
=> ? = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [5,1]
=> [1]
=> ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => [4,2]
=> [2]
=> ? = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [5,1]
=> [1]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [4,2]
=> [2]
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => [4,2]
=> [2]
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,3,2,4,5,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [6,3,2,5,4,1] => [4,2]
=> [2]
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,1]
=> ? = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [5,1]
=> [1]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [5,1]
=> [1]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [5,1]
=> [1]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => [5,1]
=> [1]
=> ? = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [5,1]
=> [1]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,7,5,4,6,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,3,7,6,5,2] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,6,3,5,4,2,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,6,3,5,4,7,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,7,3,5,4,6,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,6,4,3,5,2,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,6,4,3,5,7,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,7,4,3,5,6,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,7,4,3,6,5,2] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,7,4,5,3,6,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,6,5,4] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,6,5,3] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,7,4,6,5,3] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,7,5,4,6,3] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,7,4,6,5,1] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,6,5,4,1,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,7,5,4,6,1] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,7,3,4,6,5,1] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,6,3,5,4,1,7] => [4,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001001
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 40%●distinct values known / distinct values provided: 25%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 40%●distinct values known / distinct values provided: 25%
Values
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,6,1,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [5,1,2,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,4,1,3,7,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,1,7,3,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,5,1,3,7,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,3,6,7,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,3,6,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,6,3,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,6,3,7,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,1,6,7,3,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,1,3,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,1,3,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,3,2,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,7,2,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001000
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 40%●distinct values known / distinct values provided: 25%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 40%●distinct values known / distinct values provided: 25%
Values
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,6,1,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [5,1,2,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,4,1,3,7,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,1,7,3,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,5,1,3,7,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,3,6,7,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,3,6,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,6,3,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,6,3,7,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,1,6,7,3,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,1,3,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,1,3,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,3,2,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,7,2,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001515
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 40%●distinct values known / distinct values provided: 25%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 40%●distinct values known / distinct values provided: 25%
Values
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 0 + 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 4
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 4
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 4
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 4
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 4
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 4
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,6,1,2,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 0 + 4
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [5,1,2,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 4
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 4
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1 + 4
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 4
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 4
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,4,1,3,7,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,1,7,3,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,5,1,3,7,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,3,6,7,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,3,6,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,6,3,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,6,3,7,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,1,6,7,3,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,1,3,4,7,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 0 + 4
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,1,3,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1 + 4
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 4
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 + 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 0 + 4
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,4,5,7,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,7,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,3,2,7,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,7,2,4,5,6] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 4
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1 + 4
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St000481
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 50%
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2 = 0 + 2
[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]
=> [4,2,1]
=> 2 = 0 + 2
[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]
=> [4,3,1]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2 = 0 + 2
[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]
=> [4,3,1,1]
=> 2 = 0 + 2
[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]
=> [4,3,2]
=> 2 = 0 + 2
[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]
=> [4,3,2,1]
=> 3 = 1 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 3 = 1 + 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 2 = 0 + 2
[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]
=> [5,3,2,1]
=> 3 = 1 + 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> ? = 0 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> 2 = 0 + 2
[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]
=> [5,4,2,1]
=> 3 = 1 + 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? = 0 + 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 0 + 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 0 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> 2 = 0 + 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 0 + 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 0 + 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 2 = 0 + 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 1 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 3 = 1 + 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 3 = 1 + 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> ? = 1 + 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 0 + 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 2 + 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,3,1]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,1,1,1]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2,1]
=> ? = 0 + 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,1,1,1]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1,1]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> ? = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2]
=> ? = 0 + 2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2]
=> ? = 0 + 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2]
=> ? = 0 + 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1]
=> ? = 1 + 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,1,1]
=> ? = 1 + 2
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1,1]
=> ? = 1 + 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,1,1]
=> ? = 0 + 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> ? = 2 + 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,1]
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1]
=> ? = 0 + 2
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> ? = 0 + 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 0 + 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> ? = 0 + 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> ? = 1 + 2
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,1]
=> ? = 0 + 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,1,1,1]
=> ? = 0 + 2
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> ? = 0 + 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1,1]
=> ? = 0 + 2
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,1,1]
=> ? = 0 + 2
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 0 + 2
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,3,1,1,1]
=> ? = 0 + 2
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? = 0 + 2
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1]
=> ? = 0 + 2
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? = 0 + 2
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2]
=> ? = 0 + 2
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2]
=> ? = 0 + 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 0 + 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? = 1 + 2
Description
The number of upper covers of a partition in dominance order.
The following 205 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000479The Ramsey number of a graph. St000054The first entry of the permutation. St000264The girth of a graph, which is not a tree. St001645The pebbling number of a connected graph. St000862The number of parts of the shifted shape of a permutation. St001549The number of restricted non-inversions between exceedances. St000007The number of saliances of the permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000617The number of global maxima of a Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000546The number of global descents of a permutation. St000268The number of strongly connected orientations of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001330The hat guessing number of a graph. St001654The monophonic hull number of a graph. St001716The 1-improper chromatic number of a graph. St001116The game chromatic number of a graph. St001427The number of descents of a signed permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001963The tree-depth of a graph. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000322The skewness of a graph. St001717The largest size of an interval in a poset. St000652The maximal difference between successive positions of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001726The number of visible inversions of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000956The maximal displacement of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001597The Frobenius rank of a skew partition. St001060The distinguishing index of a graph. St001394The genus of a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000470The number of runs in a permutation. St000297The number of leading ones in a binary word. St001260The permanent of an alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001728The number of invisible descents of a permutation. St000451The length of the longest pattern of the form k 1 2. St000288The number of ones in a binary word. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000619The number of cyclic descents of a permutation. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000929The constant term of the character polynomial of an integer partition. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001615The number of join prime elements of a lattice. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000640The rank of the largest boolean interval in a poset. St000993The multiplicity of the largest part of an integer partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001256Number of simple reflexive modules that are 2-stable reflexive. St001820The size of the image of the pop stack sorting operator. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000159The number of distinct parts of the integer partition. St000912The number of maximal antichains in a poset. St001432The order dimension of the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000916The packing number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St000258The burning number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000918The 2-limited packing number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St001829The common independence number of a graph.
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