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Your data matches 457 different statistics following compositions of up to 3 maps.
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Matching statistic: St000253
(load all 184 compositions to match this statistic)
(load all 184 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> {{1},{2}}
=> 0
[1,1,0,0]
=> {{1,2}}
=> 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> {{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: St001418
(load all 167 compositions to match this statistic)
(load all 167 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St000254
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00115: Set partitions —Kasraoui-Zeng⟶ 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},{2}}
=> {{1},{2}}
=> 0
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1},{2,5},{3},{4}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1},{2,4,5},{3}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1},{2,4},{3,5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1,4},{2},{3},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,4,5},{2},{3}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,4},{2},{3,5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 1
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: St000319
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000392
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => 01 => 1
[1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000480
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000481
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 0
[1,1,0,0]
=> [1,2] => [1,1]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000659
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[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
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000730
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
Description
The maximal arc length of a set partition.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
The following 447 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000010The length of the partition. St000147The largest part of an integer partition. St000346The number of coarsenings of a partition. St000668The least common multiple of the parts of the partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001029The size of the core of a graph. St000024The number of double up and double down steps of a Dyck path. St000143The largest repeated part of a partition. St000157The number of descents of a standard tableau. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000251The number of nonsingleton blocks of a set partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000272The treewidth of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000378The diagonal inversion number of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000658The number of rises of length 2 of a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000897The number of different multiplicities of parts of an integer partition. St000919The number of maximal left branches of a binary tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001484The number of singletons of an integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001743The discrepancy of a graph. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001931The weak major index of an integer composition regarded as a word. St000013The height of a Dyck path. St000058The order of a permutation. St000105The number of blocks in the set partition. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000381The largest part of an integer composition. St000397The Strahler number of a rooted tree. St000444The length of the maximal rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000759The smallest missing part in an integer partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001494The Alon-Tarsi number of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001580The acyclic chromatic number of a graph. St001814The number of partitions interlacing the given partition. St001963The tree-depth of a graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. 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. St000259The diameter of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001093The detour number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. 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 $eA$. St000264The girth of a graph, which is not a tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St000701The protection number of a binary tree. St000629The defect of a binary word. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000993The multiplicity of the largest part of an integer partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000845The maximal number of elements covered by an element in a poset. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000662The staircase size of the code of a permutation. St000451The length of the longest pattern of the form k 1 2. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000834The number of right outer peaks of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000485The length of the longest cycle of a permutation. St000846The maximal number of elements covering an element of a poset. St000527The width of the poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000296The length of the symmetric border of a binary word. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St000035The number of left outer peaks of a permutation. St000260The radius of a connected graph. St000884The number of isolated descents of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000660The number of rises of length at least 3 of a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St000245The number of ascents of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000651The maximal size of a rise in a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000028The number of stack-sorts needed to sort a permutation. St000352The Elizalde-Pak rank of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000094The depth of an ordered tree. 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. St000741The Colin de Verdière graph invariant. 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. St000742The number of big ascents of a permutation after prepending zero. St001060The distinguishing index of a graph. St001737The number of descents of type 2 in a permutation. St000354The number of recoils of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000862The number of parts of the shifted shape of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000141The maximum drop size of a permutation. St001625The Möbius invariant of a lattice. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001890The maximum magnitude of the Möbius function of a poset. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000456The monochromatic index of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000080The rank of the poset. St000081The number of edges of a graph. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000271The chromatic index of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000472The sum of the ascent bottoms of a permutation. St000537The cutwidth of a graph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000778The metric dimension of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001270The bandwidth of a graph. St001271The competition number of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001391The disjunction number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St001973The Gromov width of a graph. St000086The number of subgraphs. St000087The number of induced subgraphs. St000166The depth minus 1 of an ordered tree. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000343The number of spanning subgraphs of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000972The composition number of a graph. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001581The achromatic number of a graph. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St000379The number of Hamiltonian cycles in a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001981The size of the largest square of zeros in the top left corner of an alternating sign matrix. St001570The minimal number of edges to add to make a graph Hamiltonian. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000779The tier of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001353The number of prime nodes in the modular decomposition of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000092The number of outer peaks of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001471The magnitude of a Dyck path. St001734The lettericity of a graph. St000638The number of up-down runs of a permutation. St001555The order of a signed permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. 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 $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001597The Frobenius rank of a skew partition. St000630The length of the shortest palindromic decomposition of a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000764The number of strong records in an integer composition. St000920The logarithmic height of a Dyck path. St001624The breadth of a lattice. St001870The number of positive entries followed by a negative entry in a signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000632The jump number of the poset. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001730The number of times the path corresponding to a binary word crosses the base line. St001960The number of descents of a permutation minus one if its first entry is not one. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.
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