- FindStat

Your data matches 418 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000761
(load all 4 compositions to match this statistic)

Mapping

St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents in an integer composition. A composition has an ascent, or rise, at position

$i$ if

$a_i < a_{i+1}$ .

Matching statistic: St000758
(load all 12 compositions to match this statistic)

Mapping

St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the longest staircase fitting into an integer composition. For a given composition

$c_1,\dots,c_n$ , this is the maximal number

$\ell$ such that there are indices

$i_1 < \dots < i_\ell$ with

$c_{i_k} \geq k$ , see [def.3.1, 1]

Matching statistic: St000764
(load all 5 compositions to match this statistic)

Mapping

St000764: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of strong records in an integer composition. A strong record is an element

$a_i$ such that

$a_i > a_j$ for all

$j < i$ . In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.

Matching statistic: St000760
(load all 5 compositions to match this statistic)

Mapping

Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the longest strictly decreasing subsequence of parts of an integer composition. By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.

Matching statistic: St000292
(load all 11 compositions to match this statistic)

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2] => [1,1] => 11 => 0 = -1 + 1
[1,2] => [2,1] => 101 => 1 = 0 + 1
[2,1] => [1,2] => 110 => 0 = -1 + 1
[3] => [1,1,1] => 111 => 0 = -1 + 1
[1,1,2] => [3,1] => 1001 => 1 = 0 + 1
[1,2,1] => [2,2] => 1010 => 1 = 0 + 1
[1,3] => [2,1,1] => 1011 => 1 = 0 + 1
[2,1,1] => [1,3] => 1100 => 0 = -1 + 1
[3,1] => [1,1,2] => 1110 => 0 = -1 + 1
[4] => [1,1,1,1] => 1111 => 0 = -1 + 1
[1,1,1,2] => [4,1] => 10001 => 1 = 0 + 1
[1,1,2,1] => [3,2] => 10010 => 1 = 0 + 1
[1,1,3] => [3,1,1] => 10011 => 1 = 0 + 1
[1,2,1,1] => [2,3] => 10100 => 1 = 0 + 1
[1,3,1] => [2,1,2] => 10110 => 1 = 0 + 1
[1,4] => [2,1,1,1] => 10111 => 1 = 0 + 1
[2,1,1,1] => [1,4] => 11000 => 0 = -1 + 1
[3,1,1] => [1,1,3] => 11100 => 0 = -1 + 1
[4,1] => [1,1,1,2] => 11110 => 0 = -1 + 1
[5] => [1,1,1,1,1] => 11111 => 0 = -1 + 1
[1,1,1,1,2] => [5,1] => 100001 => 1 = 0 + 1
[1,1,1,2,1] => [4,2] => 100010 => 1 = 0 + 1
[1,1,1,3] => [4,1,1] => 100011 => 1 = 0 + 1
[1,1,2,1,1] => [3,3] => 100100 => 1 = 0 + 1
[1,1,3,1] => [3,1,2] => 100110 => 1 = 0 + 1
[1,1,4] => [3,1,1,1] => 100111 => 1 = 0 + 1
[1,2,1,1,1] => [2,4] => 101000 => 1 = 0 + 1
[1,3,1,1] => [2,1,3] => 101100 => 1 = 0 + 1
[1,4,1] => [2,1,1,2] => 101110 => 1 = 0 + 1
[1,5] => [2,1,1,1,1] => 101111 => 1 = 0 + 1
[2,1,1,1,1] => [1,5] => 110000 => 0 = -1 + 1
[3,1,1,1] => [1,1,4] => 111000 => 0 = -1 + 1
[4,1,1] => [1,1,1,3] => 111100 => 0 = -1 + 1
[5,1] => [1,1,1,1,2] => 111110 => 0 = -1 + 1
[6] => [1,1,1,1,1,1] => 111111 => 0 = -1 + 1
[1,1,1,1,1,2] => [6,1] => 1000001 => 1 = 0 + 1
[1,1,1,1,2,1] => [5,2] => 1000010 => 1 = 0 + 1
[1,1,1,1,3] => [5,1,1] => 1000011 => 1 = 0 + 1
[1,1,1,2,1,1] => [4,3] => 1000100 => 1 = 0 + 1
[1,1,1,3,1] => [4,1,2] => 1000110 => 1 = 0 + 1
[1,1,1,4] => [4,1,1,1] => 1000111 => 1 = 0 + 1
[1,1,2,1,1,1] => [3,4] => 1001000 => 1 = 0 + 1
[1,1,3,1,1] => [3,1,3] => 1001100 => 1 = 0 + 1
[1,1,4,1] => [3,1,1,2] => 1001110 => 1 = 0 + 1
[1,1,5] => [3,1,1,1,1] => 1001111 => 1 = 0 + 1
[1,2,1,1,1,1] => [2,5] => 1010000 => 1 = 0 + 1
[1,3,1,1,1] => [2,1,4] => 1011000 => 1 = 0 + 1
[1,4,1,1] => [2,1,1,3] => 1011100 => 1 = 0 + 1
[1,5,1] => [2,1,1,1,2] => 1011110 => 1 = 0 + 1
[1,6] => [2,1,1,1,1,1] => 1011111 => 1 = 0 + 1

Description

The number of ascents of a binary word.

Matching statistic: St000386
(load all 45 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of factors DDU in a Dyck path.

Matching statistic: St000340
(load all 12 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

$110$ and

$001$ .

Matching statistic: St000390
(load all 13 compositions to match this statistic)

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2] => [1,1] => 11 => 1 = -1 + 2
[1,2] => [2,1] => 101 => 2 = 0 + 2
[2,1] => [1,2] => 110 => 1 = -1 + 2
[3] => [1,1,1] => 111 => 1 = -1 + 2
[1,1,2] => [3,1] => 1001 => 2 = 0 + 2
[1,2,1] => [2,2] => 1010 => 2 = 0 + 2
[1,3] => [2,1,1] => 1011 => 2 = 0 + 2
[2,1,1] => [1,3] => 1100 => 1 = -1 + 2
[3,1] => [1,1,2] => 1110 => 1 = -1 + 2
[4] => [1,1,1,1] => 1111 => 1 = -1 + 2
[1,1,1,2] => [4,1] => 10001 => 2 = 0 + 2
[1,1,2,1] => [3,2] => 10010 => 2 = 0 + 2
[1,1,3] => [3,1,1] => 10011 => 2 = 0 + 2
[1,2,1,1] => [2,3] => 10100 => 2 = 0 + 2
[1,3,1] => [2,1,2] => 10110 => 2 = 0 + 2
[1,4] => [2,1,1,1] => 10111 => 2 = 0 + 2
[2,1,1,1] => [1,4] => 11000 => 1 = -1 + 2
[3,1,1] => [1,1,3] => 11100 => 1 = -1 + 2
[4,1] => [1,1,1,2] => 11110 => 1 = -1 + 2
[5] => [1,1,1,1,1] => 11111 => 1 = -1 + 2
[1,1,1,1,2] => [5,1] => 100001 => 2 = 0 + 2
[1,1,1,2,1] => [4,2] => 100010 => 2 = 0 + 2
[1,1,1,3] => [4,1,1] => 100011 => 2 = 0 + 2
[1,1,2,1,1] => [3,3] => 100100 => 2 = 0 + 2
[1,1,3,1] => [3,1,2] => 100110 => 2 = 0 + 2
[1,1,4] => [3,1,1,1] => 100111 => 2 = 0 + 2
[1,2,1,1,1] => [2,4] => 101000 => 2 = 0 + 2
[1,3,1,1] => [2,1,3] => 101100 => 2 = 0 + 2
[1,4,1] => [2,1,1,2] => 101110 => 2 = 0 + 2
[1,5] => [2,1,1,1,1] => 101111 => 2 = 0 + 2
[2,1,1,1,1] => [1,5] => 110000 => 1 = -1 + 2
[3,1,1,1] => [1,1,4] => 111000 => 1 = -1 + 2
[4,1,1] => [1,1,1,3] => 111100 => 1 = -1 + 2
[5,1] => [1,1,1,1,2] => 111110 => 1 = -1 + 2
[6] => [1,1,1,1,1,1] => 111111 => 1 = -1 + 2
[1,1,1,1,1,2] => [6,1] => 1000001 => 2 = 0 + 2
[1,1,1,1,2,1] => [5,2] => 1000010 => 2 = 0 + 2
[1,1,1,1,3] => [5,1,1] => 1000011 => 2 = 0 + 2
[1,1,1,2,1,1] => [4,3] => 1000100 => 2 = 0 + 2
[1,1,1,3,1] => [4,1,2] => 1000110 => 2 = 0 + 2
[1,1,1,4] => [4,1,1,1] => 1000111 => 2 = 0 + 2
[1,1,2,1,1,1] => [3,4] => 1001000 => 2 = 0 + 2
[1,1,3,1,1] => [3,1,3] => 1001100 => 2 = 0 + 2
[1,1,4,1] => [3,1,1,2] => 1001110 => 2 = 0 + 2
[1,1,5] => [3,1,1,1,1] => 1001111 => 2 = 0 + 2
[1,2,1,1,1,1] => [2,5] => 1010000 => 2 = 0 + 2
[1,3,1,1,1] => [2,1,4] => 1011000 => 2 = 0 + 2
[1,4,1,1] => [2,1,1,3] => 1011100 => 2 = 0 + 2
[1,5,1] => [2,1,1,1,2] => 1011110 => 2 = 0 + 2
[1,6] => [2,1,1,1,1,1] => 1011111 => 2 = 0 + 2

Description

The number of runs of ones in a binary word.

Matching statistic: St000691
(load all 6 compositions to match this statistic)

Mapping

Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2] => [2] => 10 => 1 = -1 + 2
[1,2] => [2,1] => 101 => 2 = 0 + 2
[2,1] => [1,2] => 110 => 1 = -1 + 2
[3] => [3] => 100 => 1 = -1 + 2
[1,1,2] => [1,2,1] => 1101 => 2 = 0 + 2
[1,2,1] => [2,1,1] => 1011 => 2 = 0 + 2
[1,3] => [3,1] => 1001 => 2 = 0 + 2
[2,1,1] => [1,1,2] => 1110 => 1 = -1 + 2
[3,1] => [1,3] => 1100 => 1 = -1 + 2
[4] => [4] => 1000 => 1 = -1 + 2
[1,1,1,2] => [1,1,2,1] => 11101 => 2 = 0 + 2
[1,1,2,1] => [1,2,1,1] => 11011 => 2 = 0 + 2
[1,1,3] => [1,3,1] => 11001 => 2 = 0 + 2
[1,2,1,1] => [2,1,1,1] => 10111 => 2 = 0 + 2
[1,3,1] => [3,1,1] => 10011 => 2 = 0 + 2
[1,4] => [4,1] => 10001 => 2 = 0 + 2
[2,1,1,1] => [1,1,1,2] => 11110 => 1 = -1 + 2
[3,1,1] => [1,1,3] => 11100 => 1 = -1 + 2
[4,1] => [1,4] => 11000 => 1 = -1 + 2
[5] => [5] => 10000 => 1 = -1 + 2
[1,1,1,1,2] => [1,1,1,2,1] => 111101 => 2 = 0 + 2
[1,1,1,2,1] => [1,1,2,1,1] => 111011 => 2 = 0 + 2
[1,1,1,3] => [1,1,3,1] => 111001 => 2 = 0 + 2
[1,1,2,1,1] => [1,2,1,1,1] => 110111 => 2 = 0 + 2
[1,1,3,1] => [1,3,1,1] => 110011 => 2 = 0 + 2
[1,1,4] => [1,4,1] => 110001 => 2 = 0 + 2
[1,2,1,1,1] => [2,1,1,1,1] => 101111 => 2 = 0 + 2
[1,3,1,1] => [3,1,1,1] => 100111 => 2 = 0 + 2
[1,4,1] => [4,1,1] => 100011 => 2 = 0 + 2
[1,5] => [5,1] => 100001 => 2 = 0 + 2
[2,1,1,1,1] => [1,1,1,1,2] => 111110 => 1 = -1 + 2
[3,1,1,1] => [1,1,1,3] => 111100 => 1 = -1 + 2
[4,1,1] => [1,1,4] => 111000 => 1 = -1 + 2
[5,1] => [1,5] => 110000 => 1 = -1 + 2
[6] => [6] => 100000 => 1 = -1 + 2
[1,1,1,1,1,2] => [1,1,1,1,2,1] => 1111101 => 2 = 0 + 2
[1,1,1,1,2,1] => [1,1,1,2,1,1] => 1111011 => 2 = 0 + 2
[1,1,1,1,3] => [1,1,1,3,1] => 1111001 => 2 = 0 + 2
[1,1,1,2,1,1] => [1,1,2,1,1,1] => 1110111 => 2 = 0 + 2
[1,1,1,3,1] => [1,1,3,1,1] => 1110011 => 2 = 0 + 2
[1,1,1,4] => [1,1,4,1] => 1110001 => 2 = 0 + 2
[1,1,2,1,1,1] => [1,2,1,1,1,1] => 1101111 => 2 = 0 + 2
[1,1,3,1,1] => [1,3,1,1,1] => 1100111 => 2 = 0 + 2
[1,1,4,1] => [1,4,1,1] => 1100011 => 2 = 0 + 2
[1,1,5] => [1,5,1] => 1100001 => 2 = 0 + 2
[1,2,1,1,1,1] => [2,1,1,1,1,1] => 1011111 => 2 = 0 + 2
[1,3,1,1,1] => [3,1,1,1,1] => 1001111 => 2 = 0 + 2
[1,4,1,1] => [4,1,1,1] => 1000111 => 2 = 0 + 2
[1,5,1] => [5,1,1] => 1000011 => 2 = 0 + 2
[1,6] => [6,1] => 1000001 => 2 = 0 + 2

Description

The number of changes of a binary word. This is the number of indices

$i$ such that

$w_i \neq w_{i+1}$ .

Matching statistic: St000439
(load all 3 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The position of the first down step of a Dyck path.

The following 408 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000455The second largest eigenvalue of a graph if it is integral. St000052The number of valleys of a Dyck path not on the x-axis. St000159The number of distinct parts of the integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000291The number of descents of a binary word. St000297The number of leading ones in a binary word. St000356The number of occurrences of the pattern 13-2. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000660The number of rises of length at least 3 of a Dyck path. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001691The number of kings in a graph. St001712The number of natural descents of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000011The number of touch points (or returns) of a Dyck path. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000142The number of even parts of a partition. St000183The side length of the Durfee square of an integer partition. St000273The domination number of a graph. St000288The number of ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000389The number of runs of ones of odd length in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000527The width of the poset. St000544The cop number of a graph. St000553The number of blocks of a graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000568The hook number of a binary tree. St000679The pruning number of an ordered tree. St000701The protection number of a binary tree. St000742The number of big ascents of a permutation after prepending zero. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000783The side length of the largest staircase partition fitting into a partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000862The number of parts of the shifted shape of a permutation. St000916The packing number of a graph. St000920The logarithmic height of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001432The order dimension of the partition. St001657The number of twos in an integer partition. St001732The number of peaks visible from the left. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001829The common independence number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000013The height of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000452The number of distinct eigenvalues of a graph. St000767The number of runs in an integer composition. St000918The 2-limited packing number of a graph. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001486The number of corners of the ribbon associated with an integer composition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000845The maximal number of elements covered by an element in a poset. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001271The competition number of a graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000025The number of initial rises of a Dyck path. St000069The number of maximal elements of a poset. St000201The number of leaf nodes in a binary tree. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001512The minimum rank of a graph. St001717The largest size of an interval in a poset. St000453The number of distinct Laplacian eigenvalues of a graph. St001093The detour number of a graph. St000204The number of internal nodes of a binary tree. St000272The treewidth of a graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000552The number of cut vertices of a graph. St000565The major index of a set partition. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000661The number of rises of length 3 of a Dyck path. St000846The maximal number of elements covering an element of a poset. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000948The chromatic discriminant of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001459The number of zero columns in the nullspace of a graph. St001689The number of celebrities in a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001842The major index of a set partition. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000157The number of descents of a standard tableau. St000172The Grundy number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000442The maximal area to the right of an up step of a Dyck path. St000504The cardinality of the first block of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000526The number of posets with combinatorially isomorphic order polytopes. St000678The number of up steps after the last double rise of a Dyck path. St000717The number of ordinal summands of a poset. St000874The position of the last double rise in a Dyck path. St000906The length of the shortest maximal chain in a poset. St000971The smallest closer of a set partition. St001029The size of the core of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001261The Castelnuovo-Mumford regularity 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. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001463The number of distinct columns in the nullspace of a graph. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001963The tree-depth of a graph. St000444The length of the maximal rise of a Dyck path. St000643The size of the largest orbit of antichains under Panyushev complementation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001809The index of the step at the first peak of maximal height in a 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000315The number of isolated vertices of a graph. St000647The number of big descents of a permutation. St000287The number of connected components of a graph. St000670The reversal length of a permutation. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001734The lettericity of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000080The rank of the poset. St000096The number of spanning trees of a graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000617The number of global maxima of a Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000306The bounce count of a Dyck path. St001330The hat guessing number of a graph. St001812The biclique partition number of a graph. St000822The Hadwiger number of the graph. St000919The number of maximal left branches of a binary tree. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St000796The stat' of a permutation. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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: St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001668The number of points of the poset minus the width of the poset. St000007The number of saliances of the permutation. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000988The orbit size of a permutation under Foata's bijection. St000475The number of parts equal to 1 in a partition. St000486The number of cycles of length at least 3 of a permutation. St000711The number of big exceedences of a permutation. St000836The number of descents of distance 2 of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000054The first entry of the permutation. St000495The number of inversions of distance at most 2 of a permutation. St000662The staircase size of the code of a permutation. St000831The number of indices that are either descents or recoils. St001081The number of minimal length factorizations of a permutation into star transpositions. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000149The number of cells of the partition whose leg is zero and arm is odd. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. St000260The radius of a connected graph. St000646The number of big ascents of a permutation. St000779The tier of a permutation. St000619The number of cyclic descents of a permutation. St000990The first ascent of a permutation. St001735The number of permutations with the same set of runs. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000360The number of occurrences of the pattern 32-1. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000015The number of peaks of a Dyck path. St001204Call 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000023The number of inner peaks of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000259The diameter of a connected graph. St000659The number of rises of length at least 2 of a Dyck path. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000699The toughness times the least common multiple of 1,. St001586The number of odd parts smaller than the largest even part in an integer partition. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000237The number of small exceedances. St000358The number of occurrences of the pattern 31-2. St000523The number of 2-protected nodes of a rooted tree. St000781The number of proper colouring schemes of a Ferrers diagram. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000021The number of descents 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. St000238The number of indices that are not small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000654The first descent of a permutation. St000740The last entry of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000917The open packing number of a graph. St000991The number of right-to-left minima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001375The pancake length of a permutation. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000325The width of the tree associated to a permutation. St000397The Strahler number of a rooted tree. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000638The number of up-down runs of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001346The number of parking functions that give the same permutation. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000035The number of left outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001394The genus of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001571The Cartan determinant of the integer partition. St000245The number of ascents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000663The number of right floats of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition. St000264The girth of a graph, which is not a tree. St000478Another weight of a partition according to Alladi. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000934The 2-degree of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001651The Frankl number of a lattice. St000039The number of crossings of a permutation. St000100The number of linear extensions of a poset. St000252The number of nodes of degree 3 of a binary tree. St000338The number of pixed points of a permutation. St000461The rix statistic of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001730The number of times the path corresponding to a binary word crosses the base line. St001890The maximum magnitude of the Möbius function of a poset. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001948The number of augmented double ascents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000710The number of big deficiencies of a permutation. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001388The number of non-attacking neighbors 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. St001665The number of pure excedances of a permutation. St001060The distinguishing index of a graph. 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

$eA$ . St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000756The sum of the positions of the left to right maxima of a permutation. St000456The monochromatic index of a connected graph. St001435The number of missing boxes in the first row. St001488The number of corners of a skew partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000807The sum of the heights of the valleys of the associated bargraph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000295The length of the border of a binary word. St000805The number of peaks of the associated bargraph. St001399The distinguishing number of a poset. St001875The number of simple modules with projective dimension at most 1. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.