- FindStat

Your data matches 168 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

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

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

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

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

Description

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

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

Description

The maximal multiplicity of an eigenvalue in a graph.

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

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: St000319
(load all 3 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%

Values

[1] => [1]
 => []
 => ? = 0
[1,2] => [1,1]
 => [1]
 => 0
[2,1] => [2]
 => []
 => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,3,2] => [2,1]
 => [1]
 => 0
[2,1,3] => [2,1]
 => [1]
 => 0
[2,3,1] => [3]
 => []
 => ? ∊ {0,1}
[3,1,2] => [3]
 => []
 => ? ∊ {0,1}
[3,2,1] => [2,1]
 => [1]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => [3,1]
 => [1]
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3] => [2,2]
 => [2]
 => 1
[2,3,1,4] => [3,1]
 => [1]
 => 0
[2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,1,1,1,2}
[2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,1,1,1,2}
[2,4,3,1] => [3,1]
 => [1]
 => 0
[3,1,2,4] => [3,1]
 => [1]
 => 0
[3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,1,1,1,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,2,4,1] => [3,1]
 => [1]
 => 0
[3,4,1,2] => [2,2]
 => [2]
 => 1
[3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,1,1,1,2}
[4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,1,1,1,2}
[4,1,3,2] => [3,1]
 => [1]
 => 0
[4,2,1,3] => [3,1]
 => [1]
 => 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,1,1,1,2}
[4,3,2,1] => [2,2]
 => [2]
 => 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 0
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 0
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 0
[1,3,4,5,2] => [4,1]
 => [1]
 => 0
[1,3,5,2,4] => [4,1]
 => [1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 0
[1,4,2,5,3] => [4,1]
 => [1]
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 0
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => 0
[1,5,2,3,4] => [4,1]
 => [1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 0
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => 0
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,5,4,2,3] => [4,1]
 => [1]
 => 0
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 1
[2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,4,1,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,4,5,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,5,1,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,1,4,5,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,1,5,2,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,4,2,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,4,5,1,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,1,2,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,3,1,5,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,3,5,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,5,1,2,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[4,5,2,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,1,2,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,1,4,2,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,3,1,2,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,3,4,1,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,4,1,3,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[5,4,2,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3}
[2,3,4,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,3,4,6,1,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,1,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,3,5,6,4,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,3,6,1,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,3,6,5,1,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,5,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,4,1,6,3,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,4,5,3,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,4,5,6,1,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,4,6,3,1,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,4,6,5,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,5,1,3,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,5,1,6,4,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,5,4,1,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}
[2,5,4,6,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4}

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

The following 158 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000480The number of lower covers of a partition in dominance order. St001176The size of a partition minus its first part. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001587Half of the largest even part of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000376The bounce deficit of a Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000941The number of characters of the symmetric group whose value on the partition is even. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000934The 2-degree of an integer partition. St000731The number of double exceedences of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000137The Grundy value of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001657The number of twos in an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St000929The constant term of the character polynomial of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000260The radius of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001651The Frankl number of a lattice. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001330The hat guessing number of a graph. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St000359The number of occurrences of the pattern 23-1. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001432The order dimension of the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001964The interval resolution global dimension of a poset. St001571The Cartan determinant of the integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. 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. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001487The number of inner corners of a skew partition. St001561The value of the elementary symmetric function evaluated at 1. 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

$eA$ . St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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. St001633The number of simple modules with projective dimension two in the incidence algebra of the 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. 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. St001472The permanent of the Coxeter matrix of the 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. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001867The number of alignments of type EN of a signed permutation. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001927Sparre Andersen's number of positives of a signed permutation.