- FindStat

Your data matches 332 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000667

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1

Description

The greatest common divisor of the parts of the partition.

Matching statistic: St001571

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1

Description

The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.

Matching statistic: St001392

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0 = 1 - 1

Description

The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.

Matching statistic: St001137
(load all 2 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 99%●distinct values known / distinct values provided: 60%

Values

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

Description

Number of simple modules that are 3-regular in the corresponding Nakayama algebra.

Matching statistic: St001031
(load all 3 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 82%●distinct values known / distinct values provided: 60%

Values

[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1

Description

The height of the bicoloured Motzkin path associated with the Dyck path.

Matching statistic: St001498
(load all 2 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 60%

Values

[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 1

Description

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.

Matching statistic: St000758

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 40% ●values known / values provided: 51%●distinct values known / distinct values provided: 40%

Values

[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1,2,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [2,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1,2,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [2,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1,2,1,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1,2,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [2,2,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1,2,1,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [2,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1,2,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [2,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [1,2,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [2,2,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [1,2,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [2,2,1] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1,2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1,2,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [1,2,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1,2,2,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [1,3,3] => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [2,2,2] => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [2,2,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1,2,1,1,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1,2,1,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [2,2,1,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1,2,1,1,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [2,2,1,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1,2,2,1,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [2,2,1,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1,2,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [2,1,2,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [1,2,2,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [2,1,2,1] => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => [1,3,3,3] => ? = 2 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [[2,2,2,2,2,2],[1,1,1,1]]
 => ? => ? = 1 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [[3,3,3,3,3],[2,2,2]]
 => ? => ? = 2 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [[3,3,3,3,3],[2,2]]
 => ? => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[4,3,3,3],[2]]
 => [2,3,3,3] => ? = 2 + 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[4,3,3,3],[2]]
 => [2,3,3,3] => ? = 2 + 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[4,3,3,3],[2]]
 => [2,3,3,3] => ? = 2 + 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[4,3,3,3],[2]]
 => [2,3,3,3] => ? = 2 + 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[4,3,3,3],[2]]
 => [2,3,3,3] => ? = 2 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => [1,4,4,4] => ? = 3 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[4,3,3,3],[2]]
 => [2,3,3,3] => ? = 2 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[4,4,4,2],[1,1,1]]
 => [3,3,3,2] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1,1,1],[1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1,1,1],[1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1,1,1],[1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1,1,1],[1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1,1,1],[1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1,1,1],[1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1,1,1],[1,1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1,1,1],[1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1,1,1],[2,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1,1,1],[1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1,1,1],[1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1,1,1],[1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1,1,1],[2,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1,1,1],[1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1,1,1],[1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1,1,1],[2,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [[3,2,2,2,1,1],[1,1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2,1,1],[2,1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [[3,2,2,2,1,1],[1,1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [[3,3,3,2,1,1],[2,2,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2,1,1],[2,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2,1,1],[2,1,1]]
 => ? => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2,1,1],[2,1]]
 => ? => ? = 1 + 1

Description

The length of the longest staircase fitting into an integer composition. For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]

Matching statistic: St001330
(load all 13 compositions to match this statistic)

Mapping

Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 50%●distinct values known / distinct values provided: 20%

Values

[1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ([(0,6),(1,2),(1,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2 = 1 + 1

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001232
(load all 8 compositions to match this statistic)

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 44%●distinct values known / distinct values provided: 20%

Values

[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 1 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 1 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 1 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 1 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 - 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[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]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 2 - 1

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001603

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 35%●distinct values known / distinct values provided: 20%

Values

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

Description

The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

The following 322 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000636The hull number of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001029The size of the core of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001638The book thickness of a graph. St001654The monophonic hull number of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001716The 1-improper chromatic number of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000710The number of big deficiencies of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001520The number of strict 3-descents. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000449The number of pairs of vertices of a graph with distance 4. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000260The radius of a connected graph. St000259The diameter of a connected graph. St000264The girth of a graph, which is not a tree. St000093The cardinality of a maximal independent set of vertices of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000322The skewness of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000296The length of the symmetric border of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000298The order dimension or Dushnik-Miller dimension of a poset. St000846The maximal number of elements covering an element of a poset. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000069The number of maximal elements of a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001820The size of the image of the pop stack sorting operator. St001615The number of join prime elements of a lattice. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St000741The Colin de Verdière graph invariant. St000623The number of occurrences of the pattern 52341 in a permutation. St000478Another weight of a partition according to Alladi. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000447The number of pairs of vertices of a graph with distance 3. St001060The distinguishing index of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001827The number of two-component spanning forests of a graph. St001871The number of triconnected components of a graph. St000842The breadth of a permutation. St001282The number of graphs with the same chromatic polynomial. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001765The number of connected components of the friends and strangers graph. St001261The Castelnuovo-Mumford regularity of a graph. St001306The number of induced paths on four vertices in a graph. St001315The dissociation number of a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001479The number of bridges of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000297The number of leading ones in a binary word. St001621The number of atoms of a lattice. St001625The Möbius invariant of a lattice. St000068The number of minimal elements in a poset. St001549The number of restricted non-inversions between exceedances. St000031The number of cycles in the cycle decomposition of a permutation. St000451The length of the longest pattern of the form k 1 2. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000007The number of saliances of the permutation. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000546The number of global descents of a permutation. St000834The number of right outer peaks of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000647The number of big descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000862The number of parts of the shifted shape of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001130The number of two successive successions in a permutation. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St000486The number of cycles of length at least 3 of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000654The first descent of a permutation. St000958The number of Bruhat factorizations of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001162The minimum jump of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000221The number of strong fixed points of a permutation. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000516The number of stretching pairs of a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000709The number of occurrences of 14-2-3 or 14-3-2. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000879The number of long braid edges in the graph of braid moves of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001434The number of negative sum pairs of a signed permutation. St001947The number of ties in a parking function. St000629The defect of a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000295The length of the border of a binary word. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000627The exponent of a binary word. St000895The number of ones on the main diagonal of an alternating sign matrix. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St000210Minimum over maximum difference of elements in cycles. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001597The Frobenius rank of a skew partition. St001728The number of invisible descents of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001260The permanent of an alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St000635The number of strictly order preserving maps of a poset into itself. St001429The number of negative entries in a signed permutation. St000234The number of global ascents of a permutation.