- FindStat

Your data matches 53 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000657
(load all 92 compositions to match this statistic)

Mapping

St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The smallest part of an integer composition.

Matching statistic: St000655
(load all 33 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].

Matching statistic: St000210
(load all 8 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Minimum over maximum difference of elements in cycles. Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ . The statistic is then the minimum of this value over all cycles in the permutation. For example, all permutations with a fixed-point has statistic value 0, and all permutations of

$[n]$ with only one cycle, has statistic value

$n-1$ .

Matching statistic: St000685
(load all 10 compositions to match this statistic)

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dominant dimension of the LNakayama algebra associated to a Dyck path. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].

Matching statistic: St000700
(load all 7 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The protection number of an ordered tree. This is the minimal distance from the root to a leaf.

Matching statistic: St001107

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

Matching statistic: St001119

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001119: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of a shortest maximal path in a graph.

Matching statistic: St001803

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau

$T$ is the skew row-strict tableau obtained by gluing two copies of

$T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals

$\max_C\big(2\ell(T) - \ell(C)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals

$0$ , if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.

Matching statistic: St000297

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => [1]
 => 10 => 1 = 0 + 1
[1,1] => [1,1]
 => [2]
 => 100 => 1 = 0 + 1
[2] => [2]
 => [1,1]
 => 110 => 2 = 1 + 1
[1,1,1] => [1,1,1]
 => [3]
 => 1000 => 1 = 0 + 1
[1,2] => [2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[2,1] => [2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[3] => [3]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
[1,1,1,1] => [1,1,1,1]
 => [4]
 => 10000 => 1 = 0 + 1
[1,1,2] => [2,1,1]
 => [3,1]
 => 10010 => 1 = 0 + 1
[1,2,1] => [2,1,1]
 => [3,1]
 => 10010 => 1 = 0 + 1
[1,3] => [3,1]
 => [2,1,1]
 => 10110 => 1 = 0 + 1
[2,1,1] => [2,1,1]
 => [3,1]
 => 10010 => 1 = 0 + 1
[2,2] => [2,2]
 => [2,2]
 => 1100 => 2 = 1 + 1
[3,1] => [3,1]
 => [2,1,1]
 => 10110 => 1 = 0 + 1
[4] => [4]
 => [1,1,1,1]
 => 11110 => 4 = 3 + 1
[1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 100000 => 1 = 0 + 1
[1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 100010 => 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 100010 => 1 = 0 + 1
[1,1,3] => [3,1,1]
 => [3,1,1]
 => 100110 => 1 = 0 + 1
[1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 100010 => 1 = 0 + 1
[1,2,2] => [2,2,1]
 => [3,2]
 => 10100 => 1 = 0 + 1
[1,3,1] => [3,1,1]
 => [3,1,1]
 => 100110 => 1 = 0 + 1
[1,4] => [4,1]
 => [2,1,1,1]
 => 101110 => 1 = 0 + 1
[2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 100010 => 1 = 0 + 1
[2,1,2] => [2,2,1]
 => [3,2]
 => 10100 => 1 = 0 + 1
[2,2,1] => [2,2,1]
 => [3,2]
 => 10100 => 1 = 0 + 1
[2,3] => [3,2]
 => [2,2,1]
 => 11010 => 2 = 1 + 1
[3,1,1] => [3,1,1]
 => [3,1,1]
 => 100110 => 1 = 0 + 1
[3,2] => [3,2]
 => [2,2,1]
 => 11010 => 2 = 1 + 1
[4,1] => [4,1]
 => [2,1,1,1]
 => 101110 => 1 = 0 + 1
[5] => [5]
 => [1,1,1,1,1]
 => 111110 => 5 = 4 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [6]
 => 1000000 => 1 = 0 + 1
[1,1,1,1,2] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
 => [4,1,1]
 => 1000110 => 1 = 0 + 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
 => [4,2]
 => 100100 => 1 = 0 + 1
[1,1,3,1] => [3,1,1,1]
 => [4,1,1]
 => 1000110 => 1 = 0 + 1
[1,1,4] => [4,1,1]
 => [3,1,1,1]
 => 1001110 => 1 = 0 + 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
 => [4,2]
 => 100100 => 1 = 0 + 1
[1,2,2,1] => [2,2,1,1]
 => [4,2]
 => 100100 => 1 = 0 + 1
[1,2,3] => [3,2,1]
 => [3,2,1]
 => 101010 => 1 = 0 + 1
[1,3,1,1] => [3,1,1,1]
 => [4,1,1]
 => 1000110 => 1 = 0 + 1
[1,3,2] => [3,2,1]
 => [3,2,1]
 => 101010 => 1 = 0 + 1
[1,4,1] => [4,1,1]
 => [3,1,1,1]
 => 1001110 => 1 = 0 + 1
[1,5] => [5,1]
 => [2,1,1,1,1]
 => 1011110 => 1 = 0 + 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1 = 0 + 1
[2,1,1,2] => [2,2,1,1]
 => [4,2]
 => 100100 => 1 = 0 + 1
[2,1,2,1] => [2,2,1,1]
 => [4,2]
 => 100100 => 1 = 0 + 1

Description

The number of leading ones in a binary word.

Matching statistic: St000326
(load all 3 compositions to match this statistic)

Mapping

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => 10 => 10 => 1 = 0 + 1
[1,1] => [1,1]
 => 110 => 110 => 1 = 0 + 1
[2] => [2]
 => 100 => 010 => 2 = 1 + 1
[1,1,1] => [1,1,1]
 => 1110 => 1110 => 1 = 0 + 1
[1,2] => [2,1]
 => 1010 => 1100 => 1 = 0 + 1
[2,1] => [2,1]
 => 1010 => 1100 => 1 = 0 + 1
[3] => [3]
 => 1000 => 0010 => 3 = 2 + 1
[1,1,1,1] => [1,1,1,1]
 => 11110 => 11110 => 1 = 0 + 1
[1,1,2] => [2,1,1]
 => 10110 => 11010 => 1 = 0 + 1
[1,2,1] => [2,1,1]
 => 10110 => 11010 => 1 = 0 + 1
[1,3] => [3,1]
 => 10010 => 10100 => 1 = 0 + 1
[2,1,1] => [2,1,1]
 => 10110 => 11010 => 1 = 0 + 1
[2,2] => [2,2]
 => 1100 => 0110 => 2 = 1 + 1
[3,1] => [3,1]
 => 10010 => 10100 => 1 = 0 + 1
[4] => [4]
 => 10000 => 00010 => 4 = 3 + 1
[1,1,1,1,1] => [1,1,1,1,1]
 => 111110 => 111110 => 1 = 0 + 1
[1,1,1,2] => [2,1,1,1]
 => 101110 => 110110 => 1 = 0 + 1
[1,1,2,1] => [2,1,1,1]
 => 101110 => 110110 => 1 = 0 + 1
[1,1,3] => [3,1,1]
 => 100110 => 101010 => 1 = 0 + 1
[1,2,1,1] => [2,1,1,1]
 => 101110 => 110110 => 1 = 0 + 1
[1,2,2] => [2,2,1]
 => 11010 => 11100 => 1 = 0 + 1
[1,3,1] => [3,1,1]
 => 100110 => 101010 => 1 = 0 + 1
[1,4] => [4,1]
 => 100010 => 100100 => 1 = 0 + 1
[2,1,1,1] => [2,1,1,1]
 => 101110 => 110110 => 1 = 0 + 1
[2,1,2] => [2,2,1]
 => 11010 => 11100 => 1 = 0 + 1
[2,2,1] => [2,2,1]
 => 11010 => 11100 => 1 = 0 + 1
[2,3] => [3,2]
 => 10100 => 01100 => 2 = 1 + 1
[3,1,1] => [3,1,1]
 => 100110 => 101010 => 1 = 0 + 1
[3,2] => [3,2]
 => 10100 => 01100 => 2 = 1 + 1
[4,1] => [4,1]
 => 100010 => 100100 => 1 = 0 + 1
[5] => [5]
 => 100000 => 000010 => 5 = 4 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1111110 => 1111110 => 1 = 0 + 1
[1,1,1,1,2] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1 = 0 + 1
[1,1,1,2,1] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1 = 0 + 1
[1,1,1,3] => [3,1,1,1]
 => 1001110 => 1010110 => 1 = 0 + 1
[1,1,2,1,1] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1 = 0 + 1
[1,1,2,2] => [2,2,1,1]
 => 110110 => 111010 => 1 = 0 + 1
[1,1,3,1] => [3,1,1,1]
 => 1001110 => 1010110 => 1 = 0 + 1
[1,1,4] => [4,1,1]
 => 1000110 => 1001010 => 1 = 0 + 1
[1,2,1,1,1] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1 = 0 + 1
[1,2,1,2] => [2,2,1,1]
 => 110110 => 111010 => 1 = 0 + 1
[1,2,2,1] => [2,2,1,1]
 => 110110 => 111010 => 1 = 0 + 1
[1,2,3] => [3,2,1]
 => 101010 => 111000 => 1 = 0 + 1
[1,3,1,1] => [3,1,1,1]
 => 1001110 => 1010110 => 1 = 0 + 1
[1,3,2] => [3,2,1]
 => 101010 => 111000 => 1 = 0 + 1
[1,4,1] => [4,1,1]
 => 1000110 => 1001010 => 1 = 0 + 1
[1,5] => [5,1]
 => 1000010 => 1000100 => 1 = 0 + 1
[2,1,1,1,1] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1 = 0 + 1
[2,1,1,2] => [2,2,1,1]
 => 110110 => 111010 => 1 = 0 + 1
[2,1,2,1] => [2,2,1,1]
 => 110110 => 111010 => 1 = 0 + 1

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.

The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000382The first part of an integer composition. St000383The last part of an integer composition. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001316The domatic number of a graph. St001829The common independence number of a graph. St000487The length of the shortest cycle of a permutation. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St000617The number of global maxima of a Dyck path. St000667The greatest common divisor of the parts of the partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000990The first ascent of a permutation. St000908The length of the shortest maximal antichain in a poset. St000906The length of the shortest maximal chain in a poset. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001481The minimal height of a peak of a Dyck path. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001571The Cartan determinant of the integer partition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000310The minimal degree of a vertex of a graph. St000314The number of left-to-right-maxima of a permutation. St000654The first descent of a permutation. St000699The toughness times the least common multiple of 1,. St000090The variation of a composition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. 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

$eA$ . St000264The girth of a graph, which is not a tree. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000260The radius of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001884The number of borders of a binary word. St001330The hat guessing number of a graph. St000907The number of maximal antichains of minimal length in a poset.