- FindStat

Your data matches 165 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000019
(load all 373 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of the support of a permutation. A permutation

$\sigma$ may be written as a product

$\sigma = s_{i_1}\dots s_{i_k}$ with

$k$ minimal, where

$s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions

$i$ and

$i+1$ . The set of indices

$\{i_1,\dots,i_k\}$ is the '''support''' of

$\sigma$ and independent of the chosen way to write

$\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of

$\sigma$ of length

$n$ is the set of indices

$1 \leq i < n$ such that

$\sigma(k) < i$ for all

$k < i$ . Thus, the connectivity set is the complement of the support.

Matching statistic: St000377

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv defect of an integer partition. This is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$ .

Matching statistic: St001176

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St000010
(load all 7 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000507
(load all 2 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1] => [1]
 => [[1]]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[1,1,0,0]
 => [2] => [2]
 => [[1,2]]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [3] => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [3] => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [4] => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [4] => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [4] => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [4] => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1] => [10,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 9 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,10] => [10,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [11] => [11]
 => [[1,2,3,4,5,6,7,8,9,10,11]]
 => ? = 10 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [10,2] => [10,2]
 => [[1,2,3,4,5,6,7,8,9,10],[11,12]]
 => ? = 10 + 1

Description

The number of ascents of a standard tableau. Entry

$i$ of a standard Young tableau is an '''ascent''' if

$i+1$ appears to the right or above

$i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000734
(load all 9 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1] => [1] => [[1]]
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [2,1] => [[1],[2]]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [[1,2]]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [[1,2],[3]]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [[1,2],[3]]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => [[1,3],[2,4],[5]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => [[1,3],[2,4],[5]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,4,2,5,1] => [[1,2,4],[3],[5]]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,4,3,5,1] => [[1,2,4],[3],[5]]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,2,4,5,1] => [[1,3,4],[2],[5]]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => [[1,3],[2,4],[5]]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => [[1,3],[2,4],[5]]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,3,1,5,2] => [[1,4],[2,5],[3]]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,2,1,5,3] => [[1,4],[2,5],[3]]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => [[1,2,5],[3],[4]]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => [[1,2,5],[3],[4]]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => [[1,3,5],[2],[4]]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 5 = 4 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [7,5,6,8,3,1,2,4] => [5,1,7,6,3,2,8,4] => ?
 => ? = 6 + 1
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [6,7,3,4,5,1,2,8] => [3,6,4,1,7,5,2,8] => ?
 => ? = 7 + 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,6,1,2,8] => [3,5,4,1,7,6,2,8] => ?
 => ? = 7 + 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,1,2,6,8] => [3,1,5,4,2,7,6,8] => ?
 => ? = 7 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 9 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 9 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,2,1,3,9] => [2,1,8,7,6,5,4,3,9] => [[1,3,9],[2,4],[5],[6],[7],[8]]
 => ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [10,11,9,8,7,6,5,4,3,2,1] => [10,11,9,8,7,6,5,4,3,2,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [9,10,3,4,5,6,7,8,1,2] => [3,4,5,6,9,7,1,10,8,2] => ?
 => ? = 7 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [7,8,9,10,5,6,1,2,3,4] => [7,1,8,2,9,5,3,10,6,4] => ?
 => ? = 7 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [5,6,7,8,9,10,1,2,3,4] => [5,6,7,1,8,2,9,3,10,4] => ?
 => ? = 8 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [11,10,9,8,7,6,5,4,3,1,2] => [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [11,12,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,11,9,12,10] => ?
 => ? = 10 + 1

Description

The last entry in the first row of a standard tableau.

Matching statistic: St000394
(load all 9 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => [1,0]
 => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 0
[1,1,0,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,1,4,5,7,8,6] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,3,8,4,5,7,6,1] => [8,1,2,4,5,7,6,3] => ?
 => ? = 7
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [3,2,1,4,5,8,7,6] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => [8,2,1,5,4,6,3,7] => ?
 => ? = 7
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7] => [6,3,2,4,1,5,8,7] => ?
 => ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => [8,3,2,4,1,6,5,7] => ?
 => ? = 7
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => [8,3,2,5,4,6,1,7] => ?
 => ? = 7
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [7,6,4,3,5,2,8,1] => [8,6,4,3,5,2,1,7] => ?
 => ? = 7
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3,10,9,8,7] => ? => ?
 => ? = 7
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [2,1,8,7,6,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 7
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 6
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 7
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 6
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [10,9,8,7,6,5,4,3,2,1,12,11] => ? => ?
 => ? = 10

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Matching statistic: St000738
(load all 7 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1] => [1] => [[1]]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => [1,2] => [[1,2]]
 => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [2,1] => [[1],[2]]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [[1,3],[2]]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [[1,2],[3]]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,4,1,2] => [[1,2],[3,4]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,4,1,3] => [[1,2],[3,4]]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,5,1,2,3] => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,5,1,3,4] => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,4,1,3,5] => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [[1,2,4,5],[3]]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => [[1,3,4],[2],[5]]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,5,2,3,1] => [[1,2],[3,4],[5]]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
 => 5 = 4 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [7,8,6,5,4,3,2,1,9] => [7,8,6,5,4,3,2,1,9] => [[1,2,9],[3],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,8,9,7,6,5,4,3,2] => [8,9,7,6,5,4,3,1,2] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 9 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [8,9,7,6,5,4,3,2,1,10] => [8,9,7,6,5,4,3,2,1,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
 => ? = 8 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [6,8,7,5,4,3,2,1,9] => [8,6,7,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8,7,9,6,5,4,3,2] => [8,7,9,6,5,4,3,1,2] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,7,9,8,6,5,4,3,2] => [9,7,8,6,5,4,3,1,2] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9,10,8,7,6,5,4,3,2] => [9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
 => ? = 8 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,11,10,9,8,7,6,5,4,3,2] => [11,10,9,8,7,6,5,4,3,1,2] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 9 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,9,11,10] => [11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,10,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
 => ? = 9 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 10 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1,10] => [[1,2,3,4,5,6,7,8,10],[9]]
 => ? = 8 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,12,1,11] => [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
 => ? = 10 + 1

Description

The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].

Matching statistic: St000288
(load all 5 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] =>  =>  => ? = 0
[1,0,1,0]
 => [1,2] => 1 => 0 => 0
[1,1,0,0]
 => [2,1] => 0 => 1 => 1
[1,0,1,0,1,0]
 => [1,2,3] => 11 => 00 => 0
[1,0,1,1,0,0]
 => [1,3,2] => 10 => 01 => 1
[1,1,0,0,1,0]
 => [2,1,3] => 01 => 10 => 1
[1,1,0,1,0,0]
 => [2,3,1] => 00 => 11 => 2
[1,1,1,0,0,0]
 => [3,2,1] => 00 => 11 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 111 => 000 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 110 => 001 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 101 => 010 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 100 => 011 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 100 => 011 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 011 => 100 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 010 => 101 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 001 => 110 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 000 => 111 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 000 => 111 => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 001 => 110 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 000 => 111 => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 000 => 111 => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 000 => 111 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1111 => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1110 => 0001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1101 => 0010 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1100 => 0011 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 1100 => 0011 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1011 => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 1010 => 0101 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1001 => 0110 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1000 => 0111 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1000 => 0111 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1001 => 0110 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1000 => 0111 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1000 => 0111 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1000 => 0111 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0111 => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0110 => 1001 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0101 => 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0100 => 1011 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 0100 => 1011 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0011 => 1100 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0010 => 1101 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0001 => 1110 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0000 => 1111 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 0000 => 1111 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 0001 => 1110 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 0000 => 1111 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 0000 => 1111 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 0000 => 1111 => 4
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0011 => 1100 => 2
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,2,8,5,6,7,4,3] => ? => ? => ? = 5
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,7,6,8,5] => ? => ? => ? = 5
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,1,4,6,5,3,8,7] => ? => ? => ? = 5
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => ? => ? => ? = 6
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,3,8,4,5,7,6,1] => ? => ? => ? = 7
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [2,4,3,1,6,5,8,7] => ? => ? => ? = 5
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [2,4,3,1,6,8,7,5] => ? => ? => ? = 6
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => ? => ? => ? = 7
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => ? => ? => ? = 7
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => ? => ? => ? = 7
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,1,10,9,8,7,6,5,4,3] => 010000000 => 101111111 => ? = 8
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => 0000000000 => 1111111111 => ? = 10
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3,10,9,8,7] => 010001000 => 101110111 => ? = 7
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,3,2,1,6,5,8,7,10,9] => 000101010 => 111010101 => ? = 6
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,3,2,1,6,5,10,9,8,7] => 000101000 => 111010111 => ? = 7
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,8,7,6,5,10,9] => 000100010 => 111011101 => ? = 7
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [4,3,2,1,10,9,8,7,6,5] => 000100000 => 111011111 => ? = 8
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [6,5,4,3,2,1,10,9,8,7] => 000001000 => 111110111 => ? = 8

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000011
(load all 2 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 93%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1] => [1] => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [2] => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,3,1] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 4 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,3,3,1] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 4 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,2,4] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,2,4] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,4] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,4,2] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,4,2] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,4,2] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [3,1,1,3] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [3,1,1,3] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1] => [2,1,1,1,1,1,1,1,1,1] => ?
 => ? = 9 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => [8,2] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,8] => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [11] => [1,1,1,1,1,1,1,1,1,1,1] => ?
 => ? = 10 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [2,4,4] => [1,1,1,2,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [2,6,2] => [1,2,1,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,2,2,2] => [1,2,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,2,4] => [1,1,1,2,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 7 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4,2] => [1,2,1,1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 7 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [6,4] => [1,1,1,2,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,2,2,2,2,2] => [1,2,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 6 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [10,2] => [1,2,1,1,1,1,1,1,1,1,1] => ?
 => ? = 10 + 1

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

The following 155 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000013The height of a Dyck path. St000691The number of changes of a binary word. St000097The order of the largest clique of the graph. St000439The position of the first down step of a Dyck path. St000147The largest part of an integer partition. St000098The chromatic number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St000839The largest opener of a set partition. St000306The bounce count of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001581The achromatic number of a graph. St000246The number of non-inversions of a permutation. St000067The inversion number of the alternating sign matrix. St000012The area of a Dyck path. St000984The number of boxes below precisely one peak. St000378The diagonal inversion number of an integer partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001479The number of bridges of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000172The Grundy number of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000808The number of up steps of the associated bargraph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000676The number of odd rises of a Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000733The row containing the largest entry of a standard tableau. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000653The last descent of a permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000081The number of edges of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000957The number of Bruhat lower covers of a permutation. St000809The reduced reflection length of the permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000956The maximal displacement of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000240The number of indices that are not small excedances. St000316The number of non-left-to-right-maxima of a permutation. St000051The size of the left subtree of a binary tree. St000204The number of internal nodes of a binary tree. St000304The load of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000356The number of occurrences of the pattern 13-2. 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

$eA$ . St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001869The maximum cut size of a graph. St000015The number of peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001480The number of simple summands of the module J^2/J^3. St000216The absolute length of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap 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. St001280The number of parts of an integer partition that are at least two. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001812The biclique partition number of a graph. St001082The number of boxed occurrences of 123 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000133The "bounce" of a permutation. St001115The number of even descents of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000235The number of indices that are not cyclical small weak excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000673The number of non-fixed points of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000327The number of cover relations in a poset. St000080The rank of the poset. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001861The number of Bruhat lower covers of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001645The pebbling number of a connected graph. St001394The genus of a permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000005The bounce statistic of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St001557The number of inversions of the second entry of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001877Number of indecomposable injective modules with projective dimension 2. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.