- FindStat

Your data matches 271 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001036
(load all 35 compositions to match this statistic)

Mapping

St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001499
(load all 38 compositions to match this statistic)

Mapping

St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.

Matching statistic: St000249
(load all 2 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of singletons ([[St000247]]) plus the number of antisingletons ([[St000248]]) of a set partition.

Matching statistic: St000155
(load all 17 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exceedances (also excedences) of a permutation. This is defined as

$exc(\sigma) = \#\{ i : \sigma(i) > i \}$ . It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic

$(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here,

$den$ is the Denert index of a permutation, see [[St000156]].

Matching statistic: St000204
(load all 3 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size

$n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an

$(n+1)$ -gon.

Matching statistic: St000245
(load all 16 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of a permutation.

Matching statistic: St000672
(load all 18 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is

$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$ for some

$(r,a,b)$ . This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

Matching statistic: St000996
(load all 17 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

Matching statistic: St001298
(load all 5 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of repeated entries in the Lehmer code of a permutation. The Lehmer code of a permutation

$\pi$ is the sequence

$(v_1,\dots,v_n)$ , with

$v_i=|\{j > i: \pi(j) < \pi(i)\}$ . This statistic counts the number of distinct elements in this sequence.

Matching statistic: St000062
(load all 8 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the longest increasing subsequence of the permutation.

The following 261 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000314The number of left-to-right-maxima of a permutation. St000021The number of descents of a permutation. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000157The number of descents of a standard tableau. St000288The number of ones in a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000662The staircase size of the code of a permutation. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001489The maximum of the number of descents and the number of inverse descents. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000007The number of saliances of the permutation. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000482The (zero)-forcing number of a graph. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St000019The cardinality of the support of a permutation. St000030The sum of the descent differences of a permutations. St000214The number of adjacencies of a permutation. St000238The number of indices that are not small weak excedances. St000203The number of external nodes of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000272The treewidth of a graph. St000354The number of recoils of a permutation. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000172The Grundy number of a graph. St000702The number of weak deficiencies of a permutation. St001029The size of the core 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. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001315The dissociation number of a graph. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000159The number of distinct parts of the integer partition. St000292The number of ascents of a binary word. St000836The number of descents of distance 2 of a permutation. St000619The number of cyclic descents of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000659The number of rises of length at least 2 of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000982The length of the longest constant subword. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000646The number of big ascents of a permutation. St001488The number of corners of a skew partition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000647The number of big descents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000648The number of 2-excedences of a permutation. St000312The number of leaves in a graph. St000675The number of centered multitunnels of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001933The largest multiplicity of a part in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001568The smallest positive integer that does not appear twice in the partition. St000054The first entry of the permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000313The number of degree 2 vertices of a graph. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000993The multiplicity of the largest part of an integer partition. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000035The number of left outer peaks of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000925The number of topologically connected components of a set partition. St000024The number of double up and double down steps of a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001469The holeyness of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000260The radius of a connected graph. St000837The number of ascents of distance 2 of a permutation. St000236The number of cyclical small weak excedances. St000871The number of very big ascents of a permutation. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000822The Hadwiger number of the graph. St000105The number of blocks in the set partition. St000445The number of rises of length 1 of a Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001692The number of vertices with higher degree than the average degree in a graph. St000352The Elizalde-Pak rank of a permutation. St001668The number of points of the poset minus the width of the poset. St001812The biclique partition number of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the 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

$eA$ . 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

$eA$ . 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

$eA$ . St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000306The bounce count of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001729The number of visible descents of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000451The length of the longest pattern of the form k 1 2. St000264The girth of a graph, which is not a tree. St001427The number of descents of a signed permutation. St001083The number of boxed occurrences of 132 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St000253The crossing number of a set partition. St001394The genus of a permutation. St000167The number of leaves of an ordered tree. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. 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

$eA$ . St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001864The number of excedances of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001948The number of augmented double ascents of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001330The hat guessing number of a graph. St000824The sum of the number of descents and the number of recoils of a permutation. St001644The dimension of a graph. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000164The number of short pairs. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001343The dimension of the reduced incidence algebra of a poset. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001902The number of potential covers of a poset. St000052The number of valleys of a Dyck path not on the x-axis. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001960The number of descents of a permutation minus one if its first entry is not one. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000872The number of very big descents of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000239The number of small weak excedances. St000443The number of long tunnels of a Dyck path. St000665The number of rafts of a permutation. 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. St001368The number of vertices of maximal degree in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001964The interval resolution global dimension of a poset. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001118The acyclic chromatic index of a graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000983The length of the longest alternating subword. St001867The number of alignments of type EN of a signed permutation. St001487The number of inner corners of a skew partition. St001896The number of right descents of a signed permutations. St000779The tier of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001090The number of pop-stack-sorts needed to sort a permutation. St001060The distinguishing index of a graph. St001435The number of missing boxes in the first row. St000381The largest part of an integer composition. St001438The number of missing boxes of a skew partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St000731The number of double exceedences of a permutation. St000834The number of right outer peaks of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001470The cyclic holeyness of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001811The Castelnuovo-Mumford regularity of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St000317The cycle descent number of a permutation. St000767The number of runs in an integer composition. St000862The number of parts of the shifted shape of a permutation. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001557The number of inversions of the second entry of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St001637The number of (upper) dissectors of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition.