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Your data matches 62 different statistics following compositions of up to 3 maps.
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Matching statistic: St000253
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000254
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => {{1,3,4},{2},{5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,6,7,5,1] => {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [2,6,4,5,3,7,8,1] => {{1,2,6,7,8},{3,4,5}}
=> ? = 2
Description
The nesting number of a set partition.
This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St001031
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St000793
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => {{1,2,3}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => {{1,3,4},{2}}
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => {{1,2,4},{3}}
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => {{1,2,3,4}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => [1,2,3,4,7,5,6,8] => ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => [1,2,5,3,4,6,7,8] => ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7,8] => [1,4,2,3,5,6,7,8] => {{1},{2,4},{3},{5},{6},{7},{8}}
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,6,2,7,8] => [1,6,2,3,4,5,7,8] => {{1},{2,6},{3},{4},{5},{7},{8}}
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7,8] => [4,1,2,3,5,6,7,8] => {{1,4},{2},{3},{5},{6},{7},{8}}
=> ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => ?
=> ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7,8] => [2,4,1,3,5,6,7,8] => ?
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7,8] => {{1,3},{2,4},{5},{6},{7},{8}}
=> ? = 1 + 1
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition.
To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$.
Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row.
Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$.
This statistic is the length of the longest partition on the diagonal of the diagram.
Matching statistic: St001277
(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
Mp00160: Permutations —graph of inversions⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,6,1,8,7] => ([(0,1),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,5,7,1,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => ([(0,1),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => ([(0,7),(1,7),(2,6),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,4,2,5,6,7,8,3] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1
Description
The degeneracy of a graph.
The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
Matching statistic: St001358
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001358: Graphs ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001358: Graphs ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,6,1,8,7] => ([(0,1),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,5,7,1,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => ([(0,1),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => ([(0,7),(1,7),(2,6),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,4,2,5,6,7,8,3] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1
Description
The largest degree of a regular subgraph of a graph.
For $k > 2$, it is an NP-complete problem to determine whether a graph has a $k$-regular subgraph, see [1].
Matching statistic: St000451
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => [3,4,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => [3,2,4,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [3,4,5,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,4,5,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [3,4,2,5,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [2,3,5,1,4] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => [3,4,2,1,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [4,5,3,2,1] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => [1,4,5,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [3,2,4,5,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => [3,2,4,1,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => [2,4,1,5,3] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => [4,2,5,3,1] => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [2,4,1,3,5] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [4,3,5,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => [2,5,1,4,3] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => [2,3,4,1,6,7,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => [2,3,4,1,6,5,7] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [6,5,1,2,3,4,7] => [3,4,5,6,2,1,7] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [4,1,2,3,7,6,5] => [2,3,4,1,7,6,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => [2,3,1,5,6,7,4] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [3,1,2,6,4,5,7] => [2,3,1,5,6,4,7] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => [2,3,1,5,4,7,6] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => [3,1,2,7,6,4,5] => [2,3,1,6,7,5,4] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [3,1,2,5,4,6,7] => [2,3,1,5,4,6,7] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [6,4,1,2,3,5,7] => [3,4,5,2,6,1,7] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [7,4,6,1,2,3,5] => [4,5,6,2,7,3,1] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [5,1,2,3,6,4,7] => [2,3,4,6,1,5,7] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => [7,6,4,1,2,3,5] => [4,5,6,3,7,2,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [5,1,2,3,7,6,4] => [2,3,4,7,1,6,5] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [5,4,1,2,3,6,7] => [3,4,5,2,1,6,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => [2,3,1,4,6,7,5] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [3,1,2,4,6,5,7] => [2,3,1,4,6,5,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [3,1,2,7,5,4,6] => [2,3,1,6,5,7,4] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [3,1,2,6,4,7,5] => [2,3,1,5,7,4,6] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [3,1,2,6,5,4,7] => [2,3,1,6,5,4,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [7,5,4,1,2,3,6] => [4,5,6,3,2,7,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [6,4,1,2,3,7,5] => [3,4,5,2,7,1,6] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [7,5,1,2,3,6,4] => [3,4,5,7,2,6,1] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [3,1,2,4,5,7,6] => [2,3,1,4,5,7,6] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [3,1,2,4,7,6,5] => [2,3,1,4,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [3,1,2,7,6,5,4] => [2,3,1,7,6,5,4] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => [2,1,7,3,4,5,6] => [2,1,4,5,6,7,3] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [2,1,6,3,4,5,7] => [2,1,4,5,6,3,7] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,3,4,7,6] => [2,1,4,5,3,7,6] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => [2,1,7,6,3,4,5] => [2,1,5,6,7,4,3] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,1,5,3,4,6,7] => [2,1,4,5,3,6,7] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [2,1,7,5,3,4,6] => [2,1,5,6,4,7,3] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,6,7,3,5] => [2,1,6,3,4,7,5] => [2,1,4,5,7,3,6] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => [2,1,6,5,3,4,7] => [2,1,5,6,4,3,7] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,4,7,3,5,6] => [2,1,7,6,5,3,4] => [2,1,6,7,5,4,3] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,5,6,3,7] => [6,3,1,2,4,5,7] => [3,4,2,5,6,1,7] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,4,1,5,3,7,6] => [5,3,1,2,4,7,6] => [3,4,2,5,1,7,6] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,4,1,5,7,3,6] => [7,6,3,1,2,4,5] => [4,5,3,6,7,2,1] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => [5,3,1,2,4,6,7] => [3,4,2,5,1,6,7] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,1,6,7,3] => [4,1,2,7,3,5,6] => [2,3,5,1,6,7,4] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,5,1,6,3,7] => [4,1,2,6,3,5,7] => [2,3,5,1,6,4,7] => ? = 2 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001761
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
St001761: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => ? = 1
Description
The maximal multiplicity of a letter in a reduced word of a permutation.
For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Matching statistic: St000704
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 35%●distinct values known / distinct values provided: 25%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 35%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? = 1 - 1
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? = 0 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? = 1 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? = 1 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? = 0 - 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? = 1 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? = 1 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? = 1 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? = 1 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? = 1 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? = 1 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 1 = 2 - 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000781
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 35%●distinct values known / distinct values provided: 25%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 35%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0]
=> [3,1,2] => [3]
=> []
=> ? = 1 - 1
[1,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? = 0 - 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? = 1 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> []
=> ? = 1 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> []
=> ? = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? = 0 - 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? = 1 - 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> []
=> ? = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? = 1 - 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> []
=> ? = 1 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> []
=> ? = 1 - 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? = 1 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> []
=> ? = 1 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> []
=> ? = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> []
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> []
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> []
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> []
=> ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6]
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [5,2]
=> [2]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [5,2]
=> [2]
=> 1 = 2 - 1
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.
The following 52 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001128The exponens consonantiae of a partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000929The constant term of the character polynomial of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000884The number of isolated descents of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St000035The number of left outer peaks of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001737The number of descents of type 2 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000996The number of exclusive left-to-right maxima of a permutation. St000527The width of the poset. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000264The girth of a graph, which is not a tree. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. 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 $eA$. St000486The number of cycles of length at least 3 of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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