- FindStat

Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001012
(load all 5 compositions to match this statistic)

Mapping

St001012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000676

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].

Matching statistic: St001318
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001318: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of vertices of the largest induced subforest with the same number of connected components of a graph.

Matching statistic: St001004
(load all 11 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].

Matching statistic: St001068

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%

Values

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

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St000053

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St001499

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%

Values

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

Description

The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.

Matching statistic: St001880
(load all 4 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 71%

Values

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

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St001879
(load all 4 compositions to match this statistic)

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 71%

Values

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

Description

The 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.

Matching statistic: St000015

Mapping

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%

Values

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

Description

The number of peaks of a Dyck path.

The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000717The number of ordinal summands of a poset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.