- FindStat

Your data matches 203 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000654
(load all 63 compositions to match this statistic)

Mapping

St000654: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first descent of a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the smallest index

$0 < i \leq n$ such that

$\pi(i) > \pi(i+1)$ where one considers

$\pi(n+1)=0$ .

Matching statistic: St000990
(load all 47 compositions to match this statistic)

Mapping

St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first ascent of a permutation. For a permutation

$\pi$ , this is the smallest index such that

$\pi(i) < \pi(i+1)$ . For the first descent, see [[St000654]].

Matching statistic: St000989
(load all 62 compositions to match this statistic)

Mapping

St000989: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of final rises of a permutation. For a permutation

$\pi$ of length

$n$ , this is the maximal

$k$ such that

$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$ Equivalently, this is

$n-1$ minus the position of the last descent [[St000653]].

Matching statistic: St000326
(load all 10 compositions to match this statistic)

Mapping

Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => 2
[2,1] => 1 => 1
[1,2,3] => 00 => 3
[1,3,2] => 01 => 2
[2,1,3] => 10 => 1
[2,3,1] => 01 => 2
[3,1,2] => 10 => 1
[3,2,1] => 11 => 1
[1,2,3,4] => 000 => 4
[1,2,4,3] => 001 => 3
[1,3,2,4] => 010 => 2
[1,3,4,2] => 001 => 3
[1,4,2,3] => 010 => 2
[1,4,3,2] => 011 => 2
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 1
[2,3,1,4] => 010 => 2
[2,3,4,1] => 001 => 3
[2,4,1,3] => 010 => 2
[2,4,3,1] => 011 => 2
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 1
[3,2,1,4] => 110 => 1
[3,2,4,1] => 101 => 1
[3,4,1,2] => 010 => 2
[3,4,2,1] => 011 => 2
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 1
[4,2,1,3] => 110 => 1
[4,2,3,1] => 101 => 1
[4,3,1,2] => 110 => 1
[4,3,2,1] => 111 => 1
[1,2,3,4,5] => 0000 => 5
[1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => 0010 => 3
[1,2,4,5,3] => 0001 => 4
[1,2,5,3,4] => 0010 => 3
[1,2,5,4,3] => 0011 => 3
[1,3,2,4,5] => 0100 => 2
[1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => 0010 => 3
[1,3,4,5,2] => 0001 => 4
[1,3,5,2,4] => 0010 => 3
[1,3,5,4,2] => 0011 => 3
[1,4,2,3,5] => 0100 => 2
[1,4,2,5,3] => 0101 => 2
[1,4,3,2,5] => 0110 => 2
[1,4,3,5,2] => 0101 => 2
[1,4,5,2,3] => 0010 => 3
[1,4,5,3,2] => 0011 => 3

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St000382
(load all 14 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first part of an integer composition.

Matching statistic: St000383
(load all 18 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The last part of an integer composition.

Matching statistic: St000678
(load all 26 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of up steps after the last double rise of a Dyck path.

Matching statistic: St000745
(load all 17 compositions to match this statistic)

Mapping

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St001050
(load all 47 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => {{1},{2}}
 => 2
[2,1] => {{1,2}}
 => 1
[1,2,3] => {{1},{2},{3}}
 => 3
[1,3,2] => {{1},{2,3}}
 => 1
[2,1,3] => {{1,2},{3}}
 => 2
[2,3,1] => {{1,2,3}}
 => 1
[3,1,2] => {{1,2,3}}
 => 1
[3,2,1] => {{1,3},{2}}
 => 2
[1,2,3,4] => {{1},{2},{3},{4}}
 => 4
[1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,3,2,4] => {{1},{2,3},{4}}
 => 2
[1,3,4,2] => {{1},{2,3,4}}
 => 1
[1,4,2,3] => {{1},{2,3,4}}
 => 1
[1,4,3,2] => {{1},{2,4},{3}}
 => 2
[2,1,3,4] => {{1,2},{3},{4}}
 => 3
[2,1,4,3] => {{1,2},{3,4}}
 => 1
[2,3,1,4] => {{1,2,3},{4}}
 => 2
[2,3,4,1] => {{1,2,3,4}}
 => 1
[2,4,1,3] => {{1,2,3,4}}
 => 1
[2,4,3,1] => {{1,2,4},{3}}
 => 2
[3,1,2,4] => {{1,2,3},{4}}
 => 2
[3,1,4,2] => {{1,2,3,4}}
 => 1
[3,2,1,4] => {{1,3},{2},{4}}
 => 3
[3,2,4,1] => {{1,3,4},{2}}
 => 1
[3,4,1,2] => {{1,3},{2,4}}
 => 2
[3,4,2,1] => {{1,2,3,4}}
 => 1
[4,1,2,3] => {{1,2,3,4}}
 => 1
[4,1,3,2] => {{1,2,4},{3}}
 => 2
[4,2,1,3] => {{1,3,4},{2}}
 => 1
[4,2,3,1] => {{1,4},{2},{3}}
 => 3
[4,3,1,2] => {{1,2,3,4}}
 => 1
[4,3,2,1] => {{1,4},{2,3}}
 => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 3
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 3
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 2
[1,4,5,3,2] => {{1},{2,3,4,5}}
 => 1

Description

The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.

Matching statistic: St001051
(load all 14 compositions to match this statistic)

Mapping

Mp00151: Permutations —to cycle type⟶ Set partitions
St001051: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => {{1},{2}}
 => 2
[2,1] => {{1,2}}
 => 1
[1,2,3] => {{1},{2},{3}}
 => 3
[1,3,2] => {{1},{2,3}}
 => 2
[2,1,3] => {{1,2},{3}}
 => 2
[2,3,1] => {{1,2,3}}
 => 1
[3,1,2] => {{1,2,3}}
 => 1
[3,2,1] => {{1,3},{2}}
 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => 4
[1,2,4,3] => {{1},{2},{3,4}}
 => 2
[1,3,2,4] => {{1},{2,3},{4}}
 => 3
[1,3,4,2] => {{1},{2,3,4}}
 => 2
[1,4,2,3] => {{1},{2,3,4}}
 => 2
[1,4,3,2] => {{1},{2,4},{3}}
 => 3
[2,1,3,4] => {{1,2},{3},{4}}
 => 3
[2,1,4,3] => {{1,2},{3,4}}
 => 2
[2,3,1,4] => {{1,2,3},{4}}
 => 2
[2,3,4,1] => {{1,2,3,4}}
 => 1
[2,4,1,3] => {{1,2,3,4}}
 => 1
[2,4,3,1] => {{1,2,4},{3}}
 => 1
[3,1,2,4] => {{1,2,3},{4}}
 => 2
[3,1,4,2] => {{1,2,3,4}}
 => 1
[3,2,1,4] => {{1,3},{2},{4}}
 => 2
[3,2,4,1] => {{1,3,4},{2}}
 => 1
[3,4,1,2] => {{1,3},{2,4}}
 => 2
[3,4,2,1] => {{1,2,3,4}}
 => 1
[4,1,2,3] => {{1,2,3,4}}
 => 1
[4,1,3,2] => {{1,2,4},{3}}
 => 1
[4,2,1,3] => {{1,3,4},{2}}
 => 1
[4,2,3,1] => {{1,4},{2},{3}}
 => 1
[4,3,1,2] => {{1,2,3,4}}
 => 1
[4,3,2,1] => {{1,4},{2,3}}
 => 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 3
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 3
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 4
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => 2
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => 3
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => 3
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 4
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 3
[1,4,5,3,2] => {{1},{2,3,4,5}}
 => 2

Description

The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. The bijection between set partitions of

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

$k$ blocks and trees with

$n+1-k$ leaves is described in Theorem 1 of [1].

The following 193 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000297The number of leading ones in a binary word. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000759The smallest missing part in an integer partition. St000916The packing number of a graph. St000991The number of right-to-left minima of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001829The common independence number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000439The position of the first down step of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. 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. St000007The number of saliances of the permutation. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000335The difference of lower and upper interactions. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000717The number of ordinal summands of a poset. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000823The number of unsplittable factors of the set partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000908The length of the shortest maximal antichain in a poset. St000917The open packing number of a graph. St000971The smallest closer of a set partition. St000993The multiplicity of the largest part of an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001463The number of distinct columns in the nullspace of a graph. St001481The minimal height of a peak of a Dyck path. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St001733The number of weak left to right maxima of a Dyck path. St001828The Euler characteristic of a graph. St000051The size of the left subtree of a binary tree. St000053The number of valleys of the Dyck path. St000090The variation of a composition. St000133The "bounce" of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000258The burning number of a graph. St000331The number of upper interactions of a Dyck path. St000738The first entry in the last row of a standard tableau. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001340The cardinality of a minimal non-edge isolating set of a graph. St000914The sum of the values of the Möbius function of a poset. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000667The greatest common divisor of the parts of the partition. St000733The row containing the largest entry of a standard tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001571The Cartan determinant of the integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001732The number of peaks visible from the left. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001432The order dimension of the partition. St000181The number of connected components of the Hasse diagram for the poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000260The radius of a connected graph. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000259The diameter of a connected graph. St000456The monochromatic index of a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001128The exponens consonantiae of a partition. St000744The length of the path to the largest entry in a standard Young tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St001389The number of partitions of the same length below the given integer partition. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000706The product of the factorials of the multiplicities of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. 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. 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

$eA$ . St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St000193The row of the unique '1' in the first column of the alternating sign matrix. 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. St000338The number of pixed points of a permutation. St000732The number of double deficiencies of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000264The girth of a graph, which is not a tree. St001280The number of parts of an integer partition that are at least two. St001564The value of the forgotten symmetric functions when all variables set to 1. St001587Half of the largest even part of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001862The number of crossings of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001720The minimal length of a chain of small intervals in a lattice. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau.