Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000934
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => [2]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2]
 => [2]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [3,2]
 => [2]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [2,2,1]
 => [2,1]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [3,2]
 => [2]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [2,2,1]
 => [2,1]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [4,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => 0
Description
The 2-degree of an integer partition. For an integer partition $\lambda$, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
Matching statistic: St001866
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 8%
Values
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,4,3,1] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 0
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [2,3,4,5,1,6] => [2,3,4,5,1,6] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,3,4,6,5,1] => [2,3,4,6,5,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [2,3,4,1,6,5] => [2,3,4,1,6,5] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,3,5] => [3,4,5,1,2,6] => [3,4,5,1,2,6] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,3,4,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [2,3,4,6,1,5] => [2,3,4,6,1,5] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => [3,4,5,1,6,2] => [3,4,5,1,6,2] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => [4,5,6,1,3,2] => [4,5,6,1,3,2] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [2,3,5,4,6,1] => [2,3,5,4,6,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [2,3,5,4,1,6] => [2,3,5,4,1,6] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,4,5] => [3,4,6,5,1,2] => [3,4,6,5,1,2] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,2,6] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,2,5] => [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,2,4,6] => [1,2,5,3,6,4] => [1,2,5,3,6,4] => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,2,6,4] => [2,3,6,4,1,5] => [2,3,6,4,1,5] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,2,4,5] => [4,5,1,3,6,2] => [4,5,1,3,6,2] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,2,3,5,6] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,2,3,6,5] => [2,3,5,6,4,1] => [2,3,5,6,4,1] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3,6] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => [2,3,5,1,6,4] => [2,3,5,1,6,4] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,6,3,5] => [3,4,6,1,2,5] => [3,4,6,1,2,5] => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,5,2,3,6] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,4,5,6,2,3] => [3,4,1,6,5,2] => [3,4,1,6,5,2] => ? = 0
Description
The nesting alignments of a signed permutation. A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that * $-i < -j < -\pi(j) < -\pi(i)$, or * $-i < j \leq \pi(j) < -\pi(i)$, or * $i < j \leq \pi(j) < \pi(i)$.