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Your data matches 63 different statistics following compositions of up to 3 maps.
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Matching statistic: St001124

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

Matching statistic: St000783

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The side length of the largest staircase partition fitting into a partition. For an integer partition

$(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer

$k$ such that

$\lambda_i > k-i$ for

$i\in\{1,\dots,k\}$ . In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram. This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.

Matching statistic: St000318

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000340

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

Matching statistic: St000167

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of leaves of an ordered tree. This is the number of nodes which do not have any children.

Matching statistic: St000925

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of topologically connected components of a set partition. For example, the set partition

$\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components

$\{1,4,5,6\}$ and

$\{2,3\}$ . The number of set partitions with only one block is [[oeis:A099947]].

Matching statistic: St000053
(load all 6 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St001068
(load all 6 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St000024
(load all 5 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Matching statistic: St000105

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of blocks in the set partition. The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]]

$S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of

$\{ 1,\ldots,n\}$ into

$k$ blocks, see [1].

The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000159The number of distinct parts of the integer partition. St001432The order dimension of the partition. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000662The staircase size of the code of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000292The number of ascents of a binary word. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001712The number of natural descents of a standard Young tableau. 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

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St000725The smallest label of a leaf of the increasing binary tree associated to a permutation.