- FindStat

Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001225
(load all 2 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.

Matching statistic: St000019

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of the support of a permutation. A permutation

$\sigma$ may be written as a product

$\sigma = s_{i_1}\dots s_{i_k}$ with

$k$ minimal, where

$s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions

$i$ and

$i+1$ . The set of indices

$\{i_1,\dots,i_k\}$ is the '''support''' of

$\sigma$ and independent of the chosen way to write

$\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of

$\sigma$ of length

$n$ is the set of indices

$1 \leq i < n$ such that

$\sigma(k) < i$ for all

$k < i$ . Thus, the connectivity set is the complement of the support.

Matching statistic: St001227

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

Matching statistic: St000505

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00091: Set partitions —rotate increasing⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The biggest entry in the block containing the 1.

Matching statistic: St000740
(load all 2 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The last entry of a permutation. This statistic is undefined for the empty permutation.

Matching statistic: St001480
(load all 2 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.

Matching statistic: St000844

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000844: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of the largest block in the direct sum decomposition of a permutation. A component of a permutation

$\pi$ is a set of consecutive numbers

$\{a,a+1,\dots, b\}$ such that

$a\leq \pi(i) \leq b$ for all

$a\leq i\leq b$ . This statistic is the size of the largest component which does not properly contain another component.

Matching statistic: St001232
(load all 3 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001229

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001229: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 83%

Values

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

Description

The vector space dimension of the first extension group between the Jacobson radical J and J^2. The vector space dimension of

$Ext_A^1(J,J^2)$ .