- FindStat

Your data matches 331 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001279
(load all 20 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [2]
 => 2
[1,0,1,0]
 => [3,1,2] => [3]
 => 3
[1,1,0,0]
 => [2,3,1] => [3]
 => 3
[1,0,1,0,1,0]
 => [4,1,2,3] => [4]
 => 4
[1,0,1,1,0,0]
 => [3,1,4,2] => [4]
 => 4
[1,1,0,0,1,0]
 => [2,4,1,3] => [4]
 => 4
[1,1,0,1,0,0]
 => [4,3,1,2] => [4]
 => 4
[1,1,1,0,0,0]
 => [2,3,4,1] => [4]
 => 4
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5]
 => 5
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5]
 => 5
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5]
 => 5
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5]
 => 5
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5]
 => 5
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5]
 => 5
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => 5
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5]
 => 5
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5]
 => 5
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5]
 => 5
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5]
 => 5
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6]
 => 6
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6]
 => 6
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [6]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6]
 => 6
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [6]
 => 6
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [6]
 => 6
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => 6
[1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [6]
 => 6
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => 6
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => 6
[1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [6]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [6]
 => 6
[1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [6]
 => 6
[1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => [6]
 => 6
[1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [6]
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6]
 => 6
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [7]
 => 7

Description

The sum of the parts of an integer partition that are at least two.

Matching statistic: St000718
(load all 54 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 3
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 4
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 5
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 6
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,3,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 7

Description

The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]

Matching statistic: St000867
(load all 3 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000867: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => [2,1] => [2]
 => 3 = 2 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 4 = 3 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,1]
 => 4 = 3 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 5 = 4 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,1]
 => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,2]
 => 5 = 4 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,1]
 => 5 = 4 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,2,1]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [2,2,1]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 6 = 5 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 6 = 5 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,2,1]
 => 6 = 5 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,2,1]
 => 6 = 5 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => [2,2,1,1]
 => 7 = 6 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => [2,2,1,1]
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5] => [2,2,1,1]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,1,3,6,4] => [2,2,1,1]
 => 7 = 6 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,4,6,2,5] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,1,5,2,6,4] => [2,2,2]
 => 7 = 6 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,2,4,5] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,1,2] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,5,1,6,2,4] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,5,6,1,2,4] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,6,1,2,4,5] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,3] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,1,2,6,3,5] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,1,5,6,2,3] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,1,6,2,3,5] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,5,1,6,2,3] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,5,6,1,2,3] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,6,1,2,3,5] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1,2,3,6,4] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,1,2,6,3,4] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,1,6,2,3,4] => [2,2,1,1]
 => 7 = 6 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,1,2,3,4] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 8 = 7 + 1

Description

The sum of the hook lengths in the first row of an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition. Put differently, for a partition of size

$n$ with first parth

$\lambda_1$ , this is

$\binom{\lambda_1}{2} + n$ .

Matching statistic: St000998
(load all 501 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1,0]
 => 2
[1,0,1,0]
 => [1,0,1,0]
 => 3
[1,1,0,0]
 => [1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 4
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 5
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 6
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 6
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 6
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
[]
 => []
 => ? = 0

Description

Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001240
(load all 501 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001240: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1,0]
 => 2
[1,0,1,0]
 => [1,0,1,0]
 => 3
[1,1,0,0]
 => [1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 4
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 5
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 6
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 6
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 6
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
[]
 => []
 => ? = 0

Description

The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra

Matching statistic: St001650
(load all 646 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001650: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1,0]
 => 2
[1,0,1,0]
 => [1,0,1,0]
 => 3
[1,1,0,0]
 => [1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 4
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 4
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 5
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 5
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 5
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 5
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 6
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 6
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 6
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 6
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 6
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 6
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 6
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 6
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
[]
 => []
 => ? = 0

Description

The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000144
(load all 501 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1,0]
 => 1 = 2 - 1
[1,0,1,0]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0]
 => [1,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 6 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 6 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 6 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 6 - 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]
 => 6 = 7 - 1
[]
 => []
 => ? = 0 - 1

Description

The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form

$1^h0^h$ ) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.

Matching statistic: St000189
(load all 600 compositions to match this statistic)

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => ([(0,1)],2)
 => 2 = 3 - 1
[1,1,0,0]
 => ([],2)
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 3 = 4 - 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => ([],3)
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 4 = 5 - 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => ([],4)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 5 = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 5 = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 5 = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 5 = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => 5 = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => 5 = 6 - 1
[1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 5 = 6 - 1
[1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => 5 = 6 - 1
[1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => 5 = 6 - 1
[1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[]
 => ?
 => ? = 0 - 1

Description

The number of elements in the poset.

Matching statistic: St000288
(load all 741 compositions to match this statistic)

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => 10 => 1 = 2 - 1
[1,0,1,0]
 => 1010 => 2 = 3 - 1
[1,1,0,0]
 => 1100 => 2 = 3 - 1
[1,0,1,0,1,0]
 => 101010 => 3 = 4 - 1
[1,0,1,1,0,0]
 => 101100 => 3 = 4 - 1
[1,1,0,0,1,0]
 => 110010 => 3 = 4 - 1
[1,1,0,1,0,0]
 => 110100 => 3 = 4 - 1
[1,1,1,0,0,0]
 => 111000 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 4 = 5 - 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 5 = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 5 = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 5 = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 5 = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 5 = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 5 = 6 - 1
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 5 = 6 - 1
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 5 = 6 - 1
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 5 = 6 - 1
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => 6 = 7 - 1
[]
 =>  => ? = 0 - 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000336
(load all 439 compositions to match this statistic)

Mapping

Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [[1],[2]]
 => 1 = 2 - 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => 2 = 3 - 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 3 = 4 - 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 3 = 4 - 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 3 = 4 - 1
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 4 = 5 - 1
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => 4 = 5 - 1
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => 5 = 6 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => 5 = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => 5 = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [[1,2,3,6,7],[4,5,8,9,10]]
 => 5 = 6 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [[1,2,3,5,8],[4,6,7,9,10]]
 => 5 = 6 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [[1,2,3,5,7],[4,6,8,9,10]]
 => 5 = 6 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [[1,2,3,5,6],[4,7,8,9,10]]
 => 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,9],[5,6,7,8,10]]
 => 5 = 6 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [[1,2,3,4,8],[5,6,7,9,10]]
 => 5 = 6 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [[1,2,3,4,7],[5,6,8,9,10]]
 => 5 = 6 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [[1,2,3,4,6],[5,7,8,9,10]]
 => 5 = 6 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => 6 = 7 - 1
[]
 => ?
 => ? = 0 - 1

Description

The leg major index of a standard tableau. The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition. It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.

The following 321 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000395The sum of the heights of the peaks of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St000026The position of the first return of a Dyck path. St000167The number of leaves of an ordered tree. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000451The length of the longest pattern of the form k 1 2. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000228The size of a partition. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000501The size of the first part in the decomposition of a permutation. St000503The maximal difference between two elements in a common block. St000703The number of deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000839The largest opener of a set partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. 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. St001430The number of positive entries in a signed permutation. St001497The position of the largest weak excedence of a permutation. St001622The number of join-irreducible elements of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001778The largest greatest common divisor of an element and its image in a permutation. St000010The length of the partition. St000019The cardinality of the support of a permutation. St000051The size of the left subtree of a binary tree. St000081The number of edges of a graph. St000209Maximum difference of elements in cycles. St000316The number of non-left-to-right-maxima of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001479The number of bridges of a graph. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001958The degree of the polynomial interpolating the values of a permutation. St000058The order of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000312The number of leaves in a graph. St000505The biggest entry in the block containing the 1. St000527The width of the poset. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. 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. St001746The coalition number of a graph. St000050The depth or height of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000171The degree of the graph. St000240The number of indices that are not small excedances. St000293The number of inversions of a binary word. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000543The size of the conjugacy class of a binary word. St000548The number of different non-empty partial sums of an integer partition. St000553The number of blocks of a graph. St000626The minimal period of a binary word. St000632The jump number of the poset. St000636The hull number of a graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000728The dimension of a set partition. St000806The semiperimeter of the associated bargraph. St000863The length of the first row of the shifted shape of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001034The area of the parallelogram polyomino associated with the Dyck path. St001120The length of a longest path in a graph. St001176The size of a partition minus its first part. St001252Half the sum of the even parts of a partition. St001267The length of the Lyndon factorization of the binary word. St001342The number of vertices in the center of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001437The flex of a binary word. St001523The degree of symmetry of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000067The inversion number of the alternating sign matrix. St000133The "bounce" of a permutation. St000210Minimum over maximum difference of elements in cycles. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000921The number of internal inversions of a binary word. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000653The last descent of a permutation. St000956The maximal displacement of a permutation. St001480The number of simple summands of the module J^2/J^3. St000225Difference between largest and smallest parts in a partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000294The number of distinct factors of a binary word. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001074The number of inversions of the cyclic embedding of a permutation. St000060The greater neighbor of the maximum. St000197The number of entries equal to positive one in the alternating sign matrix. St000216The absolute length of a permutation. St000296The length of the symmetric border of a binary word. St000385The number of vertices with out-degree 1 in a binary tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000619The number of cyclic descents of a permutation. St000627The exponent of a binary word. St000652The maximal difference between successive positions of a permutation. St000809The reduced reflection length of the permutation. St000922The minimal number such that all substrings of this length are unique. St000957The number of Bruhat lower covers of a permutation. St000982The length of the longest constant subword. St001077The prefix exchange distance of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000519The largest length of a factor maximising the subword complexity. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001082The number of boxed occurrences of 123 in a permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St000924The number of topologically connected components of a perfect matching. St000840The number of closers smaller than the largest opener in a perfect matching. St000528The height of a poset. St000837The number of ascents of distance 2 of a permutation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000906The length of the shortest maximal chain in a poset. St000080The rank of the poset. St001725The harmonious chromatic number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001820The size of the image of the pop stack sorting operator. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St000778The metric dimension of a graph. St000018The number of inversions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001626The number of maximal proper sublattices of a lattice. St001623The number of doubly irreducible elements of a lattice. St001644The dimension of a graph. St001268The size of the largest ordinal summand in the poset. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000744The length of the path to the largest entry in a standard Young tableau. St001875The number of simple modules with projective dimension at most 1. St000822The Hadwiger number of the graph. St001316The domatic number of a graph. St001645The pebbling number of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000741The Colin de Verdière graph invariant. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001468The smallest fixpoint of a permutation. St000235The number of indices that are not cyclical small weak excedances. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000456The monochromatic index of a connected graph. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000029The depth of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000450The number of edges minus the number of vertices plus 2 of a graph. St000702The number of weak deficiencies of a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001429The number of negative entries in a signed permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000095The number of triangles of a graph. St000836The number of descents of distance 2 of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001706The number of closed sets in a graph. St000327The number of cover relations in a poset. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001428The number of B-inversions of a signed permutation. St001557The number of inversions of the second entry of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000691The number of changes of a binary word. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001812The biclique partition number of a graph. St000157The number of descents of a standard tableau. St000507The number of ascents of a standard tableau. St000352The Elizalde-Pak rank of a permutation. St000203The number of external nodes of a binary tree. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000238The number of indices that are not small weak excedances. St000884The number of isolated descents of a permutation. St001486The number of corners of the ribbon associated with an integer composition. St001861The number of Bruhat lower covers of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000044The number of vertices of the unicellular map given by a perfect matching. St000135The number of lucky cars of the parking function. 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. St000017The number of inversions of a standard tableau. St000013The height of a Dyck path. St000820The number of compositions obtained by rotating the composition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St001413Half the length of the longest even length palindromic prefix of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001427The number of descents of a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001524The degree of symmetry of a binary word. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000292The number of ascents of a binary word. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001424The number of distinct squares in a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001960The number of descents of a permutation minus one if its first entry is not one. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001684The reduced word complexity of a permutation. 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. St001926Sparre Andersen's position of the maximum of a signed permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001621The number of atoms of a lattice. St000568The hook number of a binary tree. St000919The number of maximal left branches of a binary tree. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000153The number of adjacent cycles of a permutation. St000237The number of small exceedances. St000662The staircase size of the code of a permutation. St000676The number of odd rises of a Dyck path. St000845The maximal number of elements covered by an element in a poset.