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Your data matches 145 different statistics following compositions of up to 3 maps.
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Matching statistic: St000118
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(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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] => [.,[[.,.],[.,.]]]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[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] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 1
[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: St001066
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 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)
(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 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: St000931
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ 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] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 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: St001176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2,1] => [2]
=> []
=> ? = 0
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0}
[3,1,2] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0}
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> 0
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> 0
[2,3,4,1] => [4,2,3,1] => [3,1]
=> [1]
=> 0
[2,4,1,3] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[3,1,2,4] => [3,2,1,4] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [4,2,3,1] => [3,1]
=> [1]
=> 0
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> 0
[3,2,4,1] => [4,2,3,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[3,4,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[4,1,2,3] => [4,2,3,1] => [3,1]
=> [1]
=> 0
[4,1,3,2] => [4,2,3,1] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[4,2,3,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[4,3,1,2] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,3,4,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,3,5,2,4] => [1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> 0
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,4,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> 0
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> 0
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,4,5,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,5,3,4,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,5,4,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,5,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,5,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,1,4,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,2,4,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,4,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,4,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,5,1,3] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,5,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,5,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,5,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,1,2,3] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,1,3,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,2,1,3] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,2,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,3,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,3,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,3,1,4] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,3,4,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,4,1,3] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,4,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,1,4,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,2,1,4] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,2,4,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,3,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,2,1,3] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,2,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,3,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,3,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,6,5,3,1] => [6,5,4,3,2,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,6,3,4,1] => [6,5,4,3,2,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[2,5,6,4,3,1] => [6,5,4,3,2,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001877
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0
[1,2] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0}
[2,1] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0}
[1,2,3] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1}
[1,3,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1}
[2,1,3] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1}
[2,3,1] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1}
[3,1,2] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1}
[3,2,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1}
[1,2,3,4] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,3,2,4] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[2,1,3,4] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[2,3,1,4] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[2,3,4,1] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[2,4,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[3,1,2,4] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[3,4,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[4,1,2,3] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[4,3,1,2] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,3,4,5] => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,5,4,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,1,5,3,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,2,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,3,1,5,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,3,2,1,5] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,3,2,5,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,1,4,3,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,2,1,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,2,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,3,2,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,4,2,1,3] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,4,3,1,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,5,4,6] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,5,6,4] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,6,4,5] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,6,5,4] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,4,3,6,5] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,2,4,6,5,3] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,5,3,6,4] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,2,5,4,3,6] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,2,5,4,6,3] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,2,5,6,4,3] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,6,3,5,4] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,6,4,3,5] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,2,6,4,5,3] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,6,5,4,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,2,4,6,5] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,2,5,4,6] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2,5,6,4] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2,6,4,5] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2,6,5,4] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,4,2,6,5] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,5,2,6,4] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,5,4,2,6] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,5,4,6,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,6,4,2,5] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,6,5,2,4] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,6,5,4,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,4,2,3,6,5] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,4,2,5,3,6] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001524
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> []
=> => ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> 10 => 0
[2,1] => [1,1,0,0]
=> []
=> => ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 110 => 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 100 => 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 10 => 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> => ? ∊ {0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> => ? ∊ {0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,1,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,1,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,1,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,1,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,1,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,1,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3}
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 1010101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 1001101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 1010011010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> 1001011010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 1000111010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 1000111010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 1010100110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 1001100110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> 1010010110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 1001010110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 1000110110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 1000110110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,5,3,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
Description
The degree of symmetry of a binary word.
For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.
Matching statistic: St000771
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [1,2] => ([],2)
=> ? = 0 + 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0} + 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0} + 1
[2,3,1] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0} + 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[2,1,4,3] => [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[3,1,4,2] => [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[3,2,4,1] => [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,4,1,2] => [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[3,4,2,1] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1} + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,2,4,5,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,3,4,5,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,4,2,5,3] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,4,3,5,2] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,5,2,3] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,4,5,3,2] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,5,3,2,4] => [1,5,3,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,5,3,4,2] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,1,3,5,4] => [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,1,4,5,3] => [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,3,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,3,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,3,1,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,3,4,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,1,4] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,3,5,4,1] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,4,1,3,5] => [2,4,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,4,1,5,3] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,3,1,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,4,3,5,1] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,5,1,3] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,4,5,3,1] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,1,3,4] => [2,5,1,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[2,5,1,4,3] => [2,5,1,4,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,3,1,4] => [1,5,3,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,5,3,4,1] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,5,4,3,1] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[3,1,2,5,4] => [3,1,2,5,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,4,2,5] => [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[3,1,4,5,2] => [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,1,5,4,2] => [2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[3,2,4,1,5] => [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[3,2,5,1,4] => [3,2,5,1,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[3,4,1,2,5] => [2,4,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[3,4,2,1,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[3,4,5,1,2] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[3,5,1,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[4,1,2,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[4,1,3,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[4,2,5,1,3] => [3,2,5,1,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
[4,3,1,2,5] => [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2} + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [2] => [1] => ([],1)
=> 1 = 0 + 1
[2,1] => [1,1] => [2] => ([],2)
=> ? = 0 + 1
[1,2,3] => [3] => [1] => ([],1)
=> 1 = 0 + 1
[1,3,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[3,1,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[3,2,1] => [1,1,1] => [3] => ([],3)
=> ? = 1 + 1
[1,2,3,4] => [4] => [1] => ([],1)
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,3,2,4] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[1,3,4,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,4,2,3] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[1,4,3,2] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[2,1,3,4] => [1,3] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1,4] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[2,3,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,4,1,3] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[2,4,3,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[3,1,2,4] => [1,3] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[3,1,4,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,1,4] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,4,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,4,1,2] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[3,4,2,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[4,1,2,3] => [1,3] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[4,1,3,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1,3] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2,3,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,3,1,2] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[1,2,3,4,5] => [5] => [1] => ([],1)
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,4,3,5] => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,4,5,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,5,3,4] => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,5,4,3] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[1,3,2,4,5] => [2,3] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,3,2,5,4] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,3,4,2,5] => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,3,4,5,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,3,5,2,4] => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[1,4,2,3,5] => [2,3] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,4,2,5,3] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4,3,5,2] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,4,5,2,3] => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,4,5,3,2] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[1,5,2,3,4] => [2,3] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,5,2,4,3] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,5,3,2,4] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,5,3,4,2] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,5,4,2,3] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,5,4,3,2] => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[2,1,3,4,5] => [1,4] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,4,3,5] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[2,1,4,5,3] => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,5,3,4] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[2,1,5,4,3] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[2,3,1,4,5] => [2,3] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,3,1,5,4] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,3,4,1,5] => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,3,4,5,1] => [4,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,3,5,1,4] => [3,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,3,5,4,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[2,4,5,3,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[2,5,4,3,1] => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,1,4,2,5] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,1,5,2,4] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,1,5,4,2] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,2,4,1,5] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,2,5,1,4] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,2,5,4,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,4,5,2,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[3,5,4,2,1] => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,1,3,2,5] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,1,5,2,3] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,1,5,3,2] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,2,3,1,5] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,2,5,1,3] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,2,5,3,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,3,5,1,2] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,3,5,2,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[4,5,3,2,1] => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,1,3,2,4] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,1,4,2,3] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,1,4,3,2] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,2,3,1,4] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,2,4,1,3] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,2,4,3,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,3,4,1,2] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,3,4,2,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[5,4,3,2,1] => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3} + 1
[1,2,3,6,5,4] => [4,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,2,4,3,5,6] => [3,3] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
[1,2,4,6,5,3] => [4,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4} + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000621
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,1}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1}
[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]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[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]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2}
[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]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,5,2,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[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]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[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
[2,1,4,5,3] => [3,2]
=> [2]
=> 1
[2,1,5,3,4] => [3,2]
=> [2]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => [3,2]
=> [2]
=> 1
[2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 0
[2,4,3,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,1,3] => [3,2]
=> [2]
=> 1
[2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,3,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 0
[2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,3,1] => [3,2]
=> [2]
=> 1
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 0
[3,1,2,5,4] => [3,2]
=> [2]
=> 1
[3,1,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 0
[3,2,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 0
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[3,4,1,5,2] => [3,2]
=> [2]
=> 1
[3,4,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,4,5,2,1] => [3,2]
=> [2]
=> 1
[3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
The following 135 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000941The number of characters of the symmetric group whose value on the partition is even. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000668The least common multiple of the parts of the partition. St000929The constant term of the character polynomial of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000366The number of double descents of a permutation. St000731The number of double exceedences of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001527The cyclic permutation representation number of an integer 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. 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. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. 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. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000732The number of double deficiencies of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000260The radius of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001651The Frankl number of a lattice. 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. St000365The number of double ascents of a permutation. St000454The largest eigenvalue of a graph if it is integral. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001128The exponens consonantiae of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000039The number of crossings of a permutation. St000317The cycle descent number of 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001964The interval resolution global dimension of a poset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000455The second largest eigenvalue of a graph if it is integral. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001875The number of simple modules with projective dimension at most 1. St001845The number of join irreducibles minus the rank of a lattice. 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. St001175The size of a partition minus the hook length of the base cell. St001561The value of the elementary symmetric function evaluated at 1. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001868The number of alignments of type NE of a signed permutation. 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. St001435The number of missing boxes in the first row. St001846The number of elements which do not have a complement in the lattice. St000527The width of the poset. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001438The number of missing boxes of a skew partition.
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