- FindStat

Your data matches 123 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000122
(load all 4 compositions to match this statistic)

Mapping

St000122: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => 0
[[.,.],.]
 => 0
[.,[.,[.,.]]]
 => 0
[.,[[.,.],.]]
 => 0
[[.,.],[.,.]]
 => 0
[[.,[.,.]],.]
 => 0
[[[.,.],.],.]
 => 0
[.,[.,[.,[.,.]]]]
 => 0
[.,[.,[[.,.],.]]]
 => 1
[.,[[.,.],[.,.]]]
 => 0
[.,[[.,[.,.]],.]]
 => 0
[.,[[[.,.],.],.]]
 => 0
[[.,.],[.,[.,.]]]
 => 0
[[.,.],[[.,.],.]]
 => 0
[[.,[.,.]],[.,.]]
 => 0
[[[.,.],.],[.,.]]
 => 0
[[.,[.,[.,.]]],.]
 => 0
[[.,[[.,.],.]],.]
 => 0
[[[.,.],[.,.]],.]
 => 0
[[[.,[.,.]],.],.]
 => 0
[[[[.,.],.],.],.]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => 0
[.,[.,[.,[[.,.],.]]]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => 0
[.,[[.,.],[[.,.],.]]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => 0
[.,[[[.,.],.],[.,.]]]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => 0
[.,[[.,[[.,.],.]],.]]
 => 0
[.,[[[.,.],[.,.]],.]]
 => 0
[.,[[[.,[.,.]],.],.]]
 => 0
[.,[[[[.,.],.],.],.]]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => 0
[[.,.],[.,[[.,.],.]]]
 => 1
[[.,.],[[.,.],[.,.]]]
 => 0
[[.,.],[[.,[.,.]],.]]
 => 0
[[.,.],[[[.,.],.],.]]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => 0
[[.,[.,.]],[[.,.],.]]
 => 0
[[[.,.],.],[.,[.,.]]]
 => 0
[[[.,.],.],[[.,.],.]]
 => 0
[[.,[.,[.,.]]],[.,.]]
 => 0
[[.,[[.,.],.]],[.,.]]
 => 0
[[[.,.],[.,.]],[.,.]]
 => 0
[[[.,[.,.]],.],[.,.]]
 => 0
[[[[.,.],.],.],[.,.]]
 => 0
[[.,[.,[.,[.,.]]]],.]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[[.,.],.]]]}}} in a binary tree. [[oeis:A086581]] counts binary trees avoiding this pattern.

Matching statistic: St000664
(load all 8 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of right ropes of a permutation. Let

$\pi$ be a permutation of length

$n$ . A raft of

$\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of

$\pi$ . See Definition 3.10 and Example 3.11 in [1].

Matching statistic: St000779
(load all 2 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The tier of a permutation. This is the number of elements

$i$ such that

$[i+1,k,i]$ is an occurrence of the pattern

$[2,3,1]$ . For example,

$[3,5,6,1,2,4]$ has tier

$2$ , with witnesses

$[3,5,2]$ (or

$[3,6,2]$ ) and

$[5,6,4]$ . According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].

Matching statistic: St001011

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[[.,.],.]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1

Description

Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000292

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [2,1] => [2,1] => 1 => 0
[[.,.],.]
 => [1,2] => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 11 => 0
[.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => 10 => 0
[[.,.],[.,.]]
 => [1,3,2] => [3,1,2] => 10 => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 10 => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 111 => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => 110 => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,3,1] => 110 => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => 110 => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,4,1] => 100 => 0
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [4,3,1,2] => 101 => 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,4,1,2] => 100 => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,4,1,3] => 100 => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [4,1,2,3] => 100 => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => 110 => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1,4] => 100 => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [3,1,2,4] => 100 => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 100 => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1111 => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => 1110 => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,3,4,2,1] => 1110 => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => 1110 => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,5,2,1] => 1100 => 0
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,4,2,3,1] => 1101 => 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [4,5,2,3,1] => 1100 => 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,5,2,4,1] => 1100 => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,4,1] => 1100 => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => 1110 => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,5,1] => 1100 => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [4,2,3,5,1] => 1100 => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,5,1] => 1100 => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,5,1] => 1000 => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [5,4,3,1,2] => 1011 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [4,5,3,1,2] => 1010 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [5,3,4,1,2] => 1010 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [4,3,5,1,2] => 1010 => 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [3,4,5,1,2] => 1000 => 0
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [5,2,4,1,3] => 1100 => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,4,5,1,3] => 1000 => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [5,4,1,2,3] => 1001 => 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [4,5,1,2,3] => 1000 => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,5,1,4] => 1100 => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,5,1,4] => 1000 => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [3,5,1,2,4] => 1000 => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,5,1,3,4] => 1000 => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 0
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1,5] => 1110 => 0

Description

The number of ascents of a binary word.

Matching statistic: St000360

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000360: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 32-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern

$32\!\!-\!\!1$ .

Matching statistic: St000375
(load all 2 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St001086
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern

$132$ , where the matched entries are all adjacent.

Matching statistic: St001175

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St001394

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The genus of a permutation. The genus

$g(\pi)$ of a permutation

$\pi\in\mathfrak S_n$ is defined via the relation

$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$ where

$\zeta = (1,2,\dots,n)$ is the long cycle and

$z(\cdot)$ is the number of cycles in the permutation.

The following 113 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

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. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001271The competition number of a graph. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000225Difference between largest and smallest parts in a partition. St000944The 3-degree of an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000929The constant term of the character polynomial of an integer partition. St000478Another weight of a partition according to Alladi. St000660The number of rises of length at least 3 of a Dyck path. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000031The number of cycles in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000655The length of the minimal rise of a Dyck path. St001964The interval resolution global dimension of a poset. St000302The determinant of the distance matrix of a connected graph. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001960The number of descents of a permutation minus one if its first entry is not one. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. 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

$eA$ . St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St001845The number of join irreducibles minus the rank of a lattice. St000068The number of minimal elements in a poset. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001490The number of connected components of a skew partition. St000842The breadth of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000629The defect of a binary word. St000383The last part of an integer composition. St001867The number of alignments of type EN of a signed permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001811The Castelnuovo-Mumford regularity of a permutation. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000968We 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

$c_{n−1}$ . St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000445The number of rises of length 1 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path.