Processing math: 61%

Your data matches 213 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001031
(load all 42 compositions to match this statistic)
Mapping
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => 0
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St000254
(load all 16 compositions to match this statistic)
Mapping
Mp00138: Dyck paths —to noncrossing partition⟶ 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}}
 => 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}}
 => 2
[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}}
 => 2
[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}}
 => 2
[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 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: St000253
(load all 18 compositions to match this statistic)
Mapping
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%
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: St001277
(load all 3 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001277: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: St001280
(load all 4 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [2]
 => 1
[1,1,0,0]
 => [2,1] => [1,1]
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [3]
 => 1
[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]
 => [3,2,1] => [1,1,1]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [4,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,2]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [4,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [4,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [3,2]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [4,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001358
(load all 2 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001358: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: St000793
(load all 2 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000793: 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 = 0 + 1
[1,1,0,0]
 => [1,2] => {{1},{2}}
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [2,3,1] => {{1,2,3}}
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,1,3] => {{1,2},{3}}
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,3,2] => {{1},{2,3}}
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [3,1,2] => {{1,3},{2}}
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => {{1},{2},{3}}
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => {{1,2,3},{4}}
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => {{1,2,4},{3}}
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => {{1},{2,3,4}}
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => {{1,3,4},{2}}
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => {{1,3},{2},{4}}
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => {{1,2,3,4},{5}}
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => {{1,2,3},{4,5}}
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => {{1,2,4},{3,5}}
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => {{1,2,4},{3},{5}}
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => {{1,2},{3,5},{4}}
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => {{1},{2,3,5},{4}}
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => {{1,3,4},{2},{5}}
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => {{1,3},{2,4,5}}
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => {{1,3,5},{2,4}}
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => {{1,3},{2,4},{5}}
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => {{1,3},{2},{4,5}}
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => {{1,3,5},{2},{4}}
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => {{1,3},{2,5},{4}}
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2 = 1 + 1
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition. To a set partition π of {1,…,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,…,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 π, 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: St000024
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [2]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => [2,1] => [1,1]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [3]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: 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]
 => [1]
 => 1
[1,1,0,0]
 => [2,1] => [2]
 => []
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3]
 => []
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => []
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,2]
 => [2]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1]
 => [2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [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]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,1]
 => 1
Description
The largest part of an integer partition.
Matching statistic: St000251
(load all 4 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000251: 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]]
 => {{1,2}}
 => 1
[1,1,0,0]
 => [2,1] => [[1],[2]]
 => {{1},{2}}
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [[1,2,3]]
 => {{1,2,3}}
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => {{1,3},{2}}
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0
[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}}
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => {{1,3,4},{2,5}}
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => {{1,3,5},{2,4}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => {{1,3,4},{2,5}}
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => {{1,3},{2,4},{5}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 1
Description
The number of nonsingleton blocks of a set partition.
The following 203 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000010The length of the partition. St000013The height of a Dyck path. St000444The length of the maximal rise of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001471The magnitude of a Dyck path. St001029The size of the core of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000668The least common multiple of the parts of the partition. St001621The number of atoms of a lattice. St001432The order dimension of the partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001928The number of non-overlapping descents in a permutation. 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. St000260The radius of a connected graph. St000527The width of the poset. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000845The maximal number of elements covered by an element in a poset. St000443The number of long tunnels of a Dyck path. 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. St000352The Elizalde-Pak rank of a permutation. St000451The length of the longest pattern of the form k 1 2. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000053The number of valleys of the Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000390The number of runs of ones in a binary word. St000675The number of centered multitunnels of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000955Number of times one has Exti(D(A),A)>0 for i>0 for the corresponding LNakayama algebra. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. 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. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001732The number of peaks visible from the left. St001884The number of borders of a binary word. St000781The number of proper colouring schemes of a Ferrers diagram. St001597The Frobenius rank of a skew partition. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000834The number of right outer peaks of a permutation. St000741The Colin de Verdière graph invariant. St001128The exponens consonantiae of a partition. St001737The number of descents of type 2 in a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000808The number of up steps of the associated bargraph. St000454The largest eigenvalue of a graph if it is integral. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000884The number of isolated descents of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St000100The number of linear extensions of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St000028The number of stack-sorts needed to sort a permutation. St001890The maximum magnitude of the Möbius function of a poset. St001115The number of even descents of a permutation. St000308The height of the tree associated to a permutation. St000456The monochromatic index of a connected graph. St001114The number of odd descents of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000387The matching number of a graph. St001188The number of simple modules S with grade inf 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. St001344The neighbouring number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St001062The maximal size of a block of a set partition. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000035The number of left outer peaks of a permutation. St001481The minimal height of a peak of a Dyck path. St001330The hat guessing number of a graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000570The Edelman-Greene number of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001208The number of connected components of the quiver of A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra A of K[x]/(x^n). St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000647The number of big descents of a permutation. St001394The genus of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000542The number of left-to-right-minima of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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). St000455The second largest eigenvalue of a graph if it is integral. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001728The number of invisible descents of a permutation. St001162The minimum jump of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001644The dimension of a graph. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001427The number of descents of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000664The number of right ropes of a permutation. St000779The tier of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001734The lettericity of a graph. St000846The maximal number of elements covering an element of a poset. St000862The number of parts of the shifted shape of a permutation. St000633The size of the automorphism group of a poset. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000660The number of rises of length at least 3 of a Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St000007The number of saliances of the permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001624The breadth of a lattice. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. 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. St000632The jump number of the poset. St000807The sum of the heights of the valleys of the associated bargraph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001470The cyclic holeyness of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001935The number of ascents in a parking function. 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. St000298The order dimension or Dushnik-Miller dimension of a poset. St000628The balance of a binary word. St000758The length of the longest staircase fitting into an integer composition. St000805The number of peaks of the associated bargraph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block.