- FindStat

Your data matches 156 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001066
(load all 9 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple reflexive modules in the corresponding Nakayama algebra.

Matching statistic: St001483
(load all 3 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.

Matching statistic: St000118
(load all 3 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree. [[oeis:A001006]] counts binary trees avoiding this pattern.

Matching statistic: St001167

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. The top of a module is the cokernel of the inclusion of the radical of the module into the module. For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.

Matching statistic: St001253

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. For the first 196 values the statistic coincides also with the number of fixed points of

$\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details. The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).

Matching statistic: St000776

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000776: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal multiplicity of an eigenvalue in a graph.

Matching statistic: St000052

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.

Matching statistic: St001172
(load all 2 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of 1-rises at odd height of a Dyck path.

Matching statistic: St000931
(load all 11 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern UUU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].

Matching statistic: St000734

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%

Values

[1] => [1]
 => []
 => []
 => ? = 1
[1,2] => [1,1]
 => [1]
 => [[1]]
 => 1
[2,1] => [2]
 => []
 => []
 => ? = 1
[1,2,3] => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,3,2] => [2,1]
 => [1]
 => [[1]]
 => 1
[2,1,3] => [2,1]
 => [1]
 => [[1]]
 => 1
[2,3,1] => [3]
 => []
 => []
 => ? ∊ {1,2}
[3,1,2] => [3]
 => []
 => []
 => ? ∊ {1,2}
[3,2,1] => [2,1]
 => [1]
 => [[1]]
 => 1
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => [[1]]
 => 1
[1,4,2,3] => [3,1]
 => [1]
 => [[1]]
 => 1
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => [[1,2]]
 => 2
[2,3,1,4] => [3,1]
 => [1]
 => [[1]]
 => 1
[2,3,4,1] => [4]
 => []
 => []
 => ? ∊ {1,1,2,2,2,3}
[2,4,1,3] => [4]
 => []
 => []
 => ? ∊ {1,1,2,2,2,3}
[2,4,3,1] => [3,1]
 => [1]
 => [[1]]
 => 1
[3,1,2,4] => [3,1]
 => [1]
 => [[1]]
 => 1
[3,1,4,2] => [4]
 => []
 => []
 => ? ∊ {1,1,2,2,2,3}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => [[1]]
 => 1
[3,4,1,2] => [2,2]
 => [2]
 => [[1,2]]
 => 2
[3,4,2,1] => [4]
 => []
 => []
 => ? ∊ {1,1,2,2,2,3}
[4,1,2,3] => [4]
 => []
 => []
 => ? ∊ {1,1,2,2,2,3}
[4,1,3,2] => [3,1]
 => [1]
 => [[1]]
 => 1
[4,2,1,3] => [3,1]
 => [1]
 => [[1]]
 => 1
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[4,3,1,2] => [4]
 => []
 => []
 => ? ∊ {1,1,2,2,2,3}
[4,3,2,1] => [2,2]
 => [2]
 => [[1,2]]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => [[1]]
 => 1
[1,3,5,2,4] => [4,1]
 => [1]
 => [[1]]
 => 1
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => [[1]]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[1,4,5,3,2] => [4,1]
 => [1]
 => [[1]]
 => 1
[1,5,2,3,4] => [4,1]
 => [1]
 => [[1]]
 => 1
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => [[1]]
 => 1
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[2,3,4,5,1] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,3,5,1,4] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,4,1,5,3] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,4,5,3,1] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,5,1,3,4] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,5,4,1,3] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,1,4,5,2] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,1,5,2,4] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,4,2,5,1] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,4,5,1,2] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,5,2,1,4] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[3,5,4,2,1] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,1,2,5,3] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,1,5,3,2] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,3,1,5,2] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,3,5,2,1] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,5,1,2,3] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[4,5,2,3,1] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[5,1,2,3,4] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[5,1,4,2,3] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[5,3,1,2,4] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[5,3,4,1,2] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[5,4,1,3,2] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[5,4,2,1,3] => [5]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4}
[2,3,4,5,6,1] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,4,6,1,5] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,1,6,4] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,5,6,4,1] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,6,1,4,5] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,3,6,5,1,4] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,5,6,3] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,1,6,3,5] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,3,6,1] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,5,6,1,3] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,6,3,1,5] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,4,6,5,3,1] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,3,6,4] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,1,6,4,3] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,4,1,6,3] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}
[2,5,4,6,3,1] => [6]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5}

Description

The last entry in the first row of a standard tableau.

The following 146 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001462The number of factors of a standard tableaux under concatenation. St000306The bounce count of a Dyck path. St000628The balance of a binary word. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000183The side length of the Durfee square of an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001432The order dimension of the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St000678The number of up steps after the last double rise of a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000744The length of the path to the largest entry in a standard Young tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St000731The number of double exceedences of a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000993The multiplicity of the largest part of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001933The largest multiplicity of a part in an integer partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001571The Cartan determinant of the integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001597The Frobenius rank of a skew partition. St000454The largest eigenvalue of a graph if it is integral. St001722The number of minimal chains with small intervals between a binary word and the top element. St000260The radius of a connected graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001330The hat guessing number of a graph. St000284The Plancherel distribution on integer partitions. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St001877Number of indecomposable injective modules with projective dimension 2. St000359The number of occurrences of the pattern 23-1. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St000837The number of ascents of distance 2 of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001875The number of simple modules with projective dimension at most 1. St001587Half of the largest even part of an integer partition. St001060The distinguishing index of a graph. 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. St001657The number of twos in an integer partition. St001964The interval resolution global dimension of a poset. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000256The number of parts from which one can substract 2 and still get an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001845The number of join irreducibles minus the rank of a lattice. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000706The product of the factorials of the multiplicities of an integer partition. St001280The number of parts of an integer partition that are at least two. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001846The number of elements which do not have a complement in the lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000527The width of the poset. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001867The number of alignments of type EN of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001960The number of descents of a permutation minus one if its first entry is not one. St001927Sparre Andersen's number of positives of a signed permutation. St001435The number of missing boxes in the first row.