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Your data matches 299 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St001198
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001198: 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]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001206: 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]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000282
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000282: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000282: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 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]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 3 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 3 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 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]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
Description
The size of the preimage of the map 'to poset' from Ordered trees to Posets.
Matching statistic: St001534
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001534: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001534: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 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]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 3 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 3 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 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]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
Description
The alternating sum of the coefficients of the Poincare polynomial of the poset cone.
For a poset $P$ on $\{1,\dots,n\}$, let $\mathcal K_P = \{\vec x\in\mathbb R^n| x_i < x_j \text{ for } i < _P j\}$. Furthermore let $\mathcal L(\mathcal A)$ be the intersection lattice of the braid arrangement $A_{n-1}$ and let $\mathcal L^{int} = \{ X \in \mathcal L(\mathcal A) | X \cap \mathcal K_P \neq \emptyset \}$.
Then the Poincare polynomial of the poset cone is $Poin(t) = \sum_{X\in\mathcal L^{int}} |\mu(0, X)| t^{codim X}$.
This statistic records its $Poin(-1)$.
Matching statistic: St001964
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 3 - 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 3 - 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 3 - 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 3 - 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 2 - 3
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 2 - 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 2 - 3
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 3 - 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 3 - 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 - 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 3 - 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 3 - 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 3 - 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 3 - 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 3
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 2 - 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 - 3
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 2 - 3
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 2 - 3
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2 - 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2 - 3
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 3 - 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 3
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2 - 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 2 - 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 2 - 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2 - 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 2 - 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 3 - 3
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 3 - 3
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000781
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1 = 3 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [4]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [4]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [2]
=> []
=> ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1 = 3 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [4]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [4]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [2]
=> []
=> ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St001934
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1 = 3 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1 = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 3 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 3 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [4]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [4]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [3]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [2]
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [2]
=> []
=> ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> ?
=> ? = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> ?
=> ? = 3 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St001195
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1 = 3 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2]
=> [1,0,1,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> [1,0]
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> [1,0]
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> [1,0]
=> ? = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> [1,0]
=> ? = 3 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? = 3 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? = 3 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3 - 2
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: St001828
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001828: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001828: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 3 - 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]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(0,1),(0,9),(1,8),(2,5),(2,9),(3,5),(3,6),(4,6),(4,7),(5,10),(6,10),(7,8),(7,10),(8,9),(9,10)],11)
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 3 - 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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,1),(0,2),(1,8),(2,8),(3,4),(3,6),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ? = 2 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 3 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 3 - 2
Description
The Euler characteristic of a graph.
The '''Euler characteristic''' $\chi$ of a topological space is the alternating sum of the dimensions of the homology groups
$$\chi(X) = \sum_{k \geq 0} (-1)^k \dim H_k(X).$$
For a finite simplicial complex, this is equal to the alternating sum $ \sum_{k\geq 0} (-1)^k f_k$ where $f_k$ the number of $k$-dimensional simplices. A (simple) graph is a simplicial complex of dimension at most one; its vertices are the 0-simplices and its edges are the 1-simplices.
For a connected graph, the Euler characteristic is equal to $1 - g$ where $g$ is the cyclomatic number.
The following 289 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000741The Colin de Verdière graph invariant. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000908The length of the shortest maximal antichain in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001779The order of promotion on the set of linear extensions of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. 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. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001902The number of potential covers of a poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000035The number of left outer peaks of a permutation. St000264The girth of a graph, which is not a tree. St000546The number of global descents of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000779The tier of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001810The number of fixed points of a permutation smaller than its largest moved point. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000455The second largest eigenvalue of a graph if it is integral. St000878The number of ones minus the number of zeros of a binary word. St000647The number of big descents of a permutation. St000665The number of rafts of a permutation. St001096The size of the overlap set of a permutation. St000617The number of global maxima of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000624The normalized sum of the minimal distances to a greater element. St000710The number of big deficiencies of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000288The number of ones in a binary word. St000007The number of saliances of the permutation. St000834The number of right outer peaks of a permutation. St000842The breadth of a permutation. St001052The length of the exterior of a permutation. 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. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000366The number of double descents of a permutation. St000402Half the size of the symmetry class of a permutation. St000627The exponent of a binary word. St000669The number of permutations obtained by switching ascents or descents of size 2. St000696The number of cycles in the breakpoint graph of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St000022The number of fixed points of a permutation. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St000629The defect of a binary word. St000648The number of 2-excedences of a permutation. St000731The number of double exceedences of a permutation. St000768The number of peaks in an integer composition. St001394The genus of a permutation. St001846The number of elements which do not have a complement in the lattice. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001403The number of vertical separators in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000308The height of the tree associated to a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000527The width of the poset. St000836The number of descents of distance 2 of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001806The upper middle entry of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000153The number of adjacent cycles of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000237The number of small exceedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000259The diameter of a connected graph. St000353The number of inner valleys of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000638The number of up-down runs of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000742The number of big ascents of a permutation after prepending zero. St000837The number of ascents of distance 2 of a permutation. St000862The number of parts of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000958The number of Bruhat factorizations of a permutation. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001114The number of odd descents of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001220The width of a permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001569The maximal modular displacement of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000028The number of stack-sorts needed to sort a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000068The number of minimal elements in a poset. St000221The number of strong fixed points of a permutation. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000245The number of ascents of a permutation. St000260The radius of a connected graph. St000352The Elizalde-Pak rank of a permutation. St000461The rix statistic of a permutation. St000649The number of 3-excedences of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000884The number of isolated descents of a permutation. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001260The permanent of an alternating sign matrix. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001638The book thickness of a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000359The number of occurrences of the pattern 23-1. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001615The number of join prime elements of a lattice. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001618The cardinality of the Frattini sublattice of a lattice. St000058The order of a permutation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000744The length of the path to the largest entry in a standard Young tableau. St001462The number of factors of a standard tableaux under concatenation. St000735The last entry on the main diagonal of a standard tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001851The number of Hecke atoms of a signed permutation. 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. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000764The number of strong records in an integer composition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000945The number of matchings in the dihedral orbit of a perfect matching. St001488The number of corners of a skew partition. St001545The second Elser number of a connected graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000295The length of the border of a binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001703The villainy of a graph. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St000454The largest eigenvalue of a graph if it is integral. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001563The value of the power-sum symmetric function evaluated at 1. St001597The Frobenius rank of a skew partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001596The number of two-by-two squares inside a skew partition. St001850The number of Hecke atoms of a permutation. St001895The oddness of a signed permutation. St000422The energy of a graph, if it is integral. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000753The Grundy value for the game of Kayles on a binary word. St001564The value of the forgotten symmetric functions when all variables set to 1. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001848The atomic length of a signed permutation. St001862The number of crossings of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000047The number of standard immaculate tableaux of a given shape.
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