- FindStat

Your data matches 145 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000480

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for

$n\neq 2$ ) can cover is

$\frac{1}{2}(\sqrt{1+8n}-3)$ and an element which covers this number of elements is given by

$(c+i,c,c-1,\dots,3,2,1)$ , where

$1\leq i\leq c+2$ .

Matching statistic: St000159
(load all 2 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Matching statistic: St000783
(load all 2 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 1 = 0 + 1
[1,2] => [1,2] => [1,1]
 => 1 = 0 + 1
[2,1] => [1,2] => [1,1]
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
 => 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,1]
 => 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[3,1,2] => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
 => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
 => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
 => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
 => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
 => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
 => 2 = 1 + 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: St001432
(load all 3 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order dimension of the partition. Given a partition

$\lambda$ , let

$I(\lambda)$ be the principal order ideal in the Young lattice generated by

$\lambda$ . The order dimension of a partition is defined as the order dimension of the poset

$I(\lambda)$ .

Matching statistic: St000318
(load all 2 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000292

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 10 => 0
[1,2] => [1,2] => [1,1]
 => 110 => 0
[2,1] => [1,2] => [1,1]
 => 110 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => 1110 => 0
[1,3,2] => [1,3,2] => [2,1]
 => 1010 => 1
[2,1,3] => [1,3,2] => [2,1]
 => 1010 => 1
[2,3,1] => [1,2,3] => [1,1,1]
 => 1110 => 0
[3,1,2] => [1,2,3] => [1,1,1]
 => 1110 => 0
[3,2,1] => [1,2,3] => [1,1,1]
 => 1110 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 10110 => 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 10110 => 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
 => 10110 => 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
 => 10110 => 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
 => 10110 => 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
 => 10110 => 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
 => 10110 => 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
 => 10110 => 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 0
[2,4,1,3] => [1,3,2,4] => [2,1,1]
 => 10110 => 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
 => 10110 => 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
 => 10110 => 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
 => 10110 => 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
 => 10110 => 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
 => 10110 => 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 0
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 0
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 0
[4,1,3,2] => [1,3,2,4] => [2,1,1]
 => 10110 => 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
 => 10110 => 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 0
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 0
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 111110 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 101110 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 101110 => 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
 => 101110 => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => 101110 => 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
 => 101110 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 101110 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 11010 => 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
 => 101110 => 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
 => 101110 => 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
 => 101110 => 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
 => 101110 => 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
 => 101110 => 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
 => 11010 => 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
 => 11010 => 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
 => 101110 => 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
 => 101110 => 1

Description

The number of ascents of a binary word.

Matching statistic: St000481

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of upper covers of a partition in dominance order.

Matching statistic: St001036

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St001037

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000291

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 10 => 1 = 0 + 1
[1,2] => [1,2] => [1,1]
 => 110 => 1 = 0 + 1
[2,1] => [1,2] => [1,1]
 => 110 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => 1110 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
 => 1010 => 2 = 1 + 1
[2,1,3] => [1,3,2] => [2,1]
 => 1010 => 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,1,1]
 => 1110 => 1 = 0 + 1
[3,1,2] => [1,2,3] => [1,1,1]
 => 1110 => 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,1,1]
 => 1110 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 10110 => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
 => 10110 => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
 => 10110 => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
 => 10110 => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
 => 10110 => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
 => 10110 => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
 => 10110 => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 111110 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 11010 => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
 => 11010 => 2 = 1 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
 => 11010 => 2 = 1 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
 => 101110 => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
 => 101110 => 2 = 1 + 1

Description

The number of descents of a binary word.

The following 135 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000390The number of runs of ones in a binary word. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000862The number of parts of the shifted shape of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000298The order dimension or Dushnik-Miller dimension of a poset. St000058The order of a permutation. St000451The length of the longest pattern of the form k 1 2. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000260The radius of a connected graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001734The lettericity of a graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001665The number of pure excedances of a permutation. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000485The length of the longest cycle of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000456The monochromatic index of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001737The number of descents of type 2 in a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001729The number of visible descents of a permutation. St000542The number of left-to-right-minima 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. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St000308The height of the tree associated to a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000397The Strahler number of a rooted tree. St000387The matching number of a graph. St000535The rank-width of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001271The competition number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001638The book thickness of a graph. St001728The number of invisible descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000328The maximum number of child nodes in a tree. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001530The depth of a Dyck path. St001644The dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001962The proper pathwidth of a graph. St000094The depth of an ordered tree. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001859The number of factors of the Stanley symmetric function associated with a permutation. St000486The number of cycles of length at least 3 of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. 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

$eA$ . St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000662The staircase size of the code of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001896The number of right descents of a signed permutations. St001624The breadth of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000454The largest eigenvalue of a graph if it is integral. St001820The size of the image of the pop stack sorting operator. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian.