- FindStat

Your data matches 243 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000126
(load all 15 compositions to match this statistic)

Mapping

St000126: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,[.,.]]]]]}}} in a binary tree. [[oeis:A036766]] counts binary trees avoiding this pattern.

Matching statistic: St001549
(load all 5 compositions to match this statistic)

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of restricted non-inversions between exceedances. This is for a permutation

$\sigma$ of length

$n$ given by

$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$

Matching statistic: St001552
(load all 6 compositions to match this statistic)

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001552: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => [1] => [1] => ? = 0
[.,[.,.]]
 => [2,1] => [1,2] => [1,2] => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[[.,.],[.,.]]
 => [3,1,2] => [1,2,3] => [1,2,3] => 0
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => [1,3,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 0
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 0
[[[[.,[.,.]],.],.],[.,[.,.]]]
 => [7,6,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,.]],.],.],[[.,.],.]]
 => [6,7,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,.]],.],[.,.]],[.,.]]
 => [7,5,2,1,3,4,6] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,.]],[.,.]],.],[.,.]]
 => [7,4,2,1,3,5,6] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,[.,.]]],.],.],[.,.]]
 => [7,3,2,1,4,5,6] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[[.,.],.]],.],.],[.,.]]
 => [7,2,3,1,4,5,6] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[[.,[.,.]],.],.],.],[.,.]]
 => [7,2,1,3,4,5,6] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,.]],.],[.,[.,.]]],.]
 => [6,5,2,1,3,4,7] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,.]],.],[[.,.],.]],.]
 => [5,6,2,1,3,4,7] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,.]],[.,.]],[.,.]],.]
 => [6,4,2,1,3,5,7] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,[.,.]]],.],[.,.]],.]
 => [6,3,2,1,4,5,7] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[[.,.],.]],.],[.,.]],.]
 => [6,2,3,1,4,5,7] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[[.,[.,.]],.],.],[.,.]],.]
 => [6,2,1,3,4,5,7] => [1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[.,[[.,[.,.]],[.,.]]],.],.]
 => [5,3,2,4,1,6,7] => [1,6,7,2,4,3,5] => [1,4,6,5,7,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[.,[[.,[.,[.,.]]],.]],.],.]
 => [4,3,2,5,1,6,7] => [1,6,7,2,5,3,4] => [1,4,6,7,5,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[.,[[.,[[.,.],.]],.]],.],.]
 => [3,4,2,5,1,6,7] => [1,6,7,2,5,3,4] => [1,4,6,7,5,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[.,[[[.,[.,.]],.],.]],.],.]
 => [3,2,4,5,1,6,7] => [1,6,7,2,4,5,3] => [1,4,7,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,.]],[.,[.,.]]],.],.]
 => [5,4,2,1,3,6,7] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,.]],[[.,.],.]],.],.]
 => [4,5,2,1,3,6,7] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,[.,.]]],[.,.]],.],.]
 => [5,3,2,1,4,6,7] => [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[[.,.],.]],[.,.]],.],.]
 => [5,2,3,1,4,6,7] => [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[[.,[.,.]],.],[.,.]],.],.]
 => [5,2,1,3,4,6,7] => [1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,[.,[.,.]]]],.],.],.]
 => [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[.,[[.,.],.]]],.],.],.]
 => [3,4,2,1,5,6,7] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[[.,.],[.,.]]],.],.],.]
 => [4,2,3,1,5,6,7] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[[.,[.,.]],.]],.],.],.]
 => [3,2,4,1,5,6,7] => [1,5,6,7,2,4,3] => [1,5,7,6,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[.,[[[.,.],.],.]],.],.],.]
 => [2,3,4,1,5,6,7] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[[.,[.,.]],[.,.]],.],.],.]
 => [4,2,1,3,5,6,7] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[[.,[.,[.,.]]],.],.],.],.]
 => [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[[.,[[.,.],.]],.],.],.],.]
 => [2,3,1,4,5,6,7] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}
[[[[[[.,[.,.]],.],.],.],.],.]
 => [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2}

Description

The number of inversions between excedances and fixed points of a permutation. This is,

$\operatorname{iefp}(\sigma) = \#\{1 \leq i,j \leq n \mid i < j = \sigma(j) < \sigma(i) \}.$

Matching statistic: St000205
(load all 3 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 92%●distinct values known / distinct values provided: 50%

Values

[.,.]
 => [1] => [1]
 => []
 => ? = 0
[.,[.,.]]
 => [2,1] => [2]
 => []
 => ? = 0
[[.,.],.]
 => [1,2] => [1,1]
 => [1]
 => 0
[.,[.,[.,.]]]
 => [3,2,1] => [2,1]
 => [1]
 => 0
[.,[[.,.],.]]
 => [2,3,1] => [3]
 => []
 => ? = 0
[[.,.],[.,.]]
 => [1,3,2] => [2,1]
 => [1]
 => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1]
 => [1]
 => 0
[[[.,.],.],.]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [2,2]
 => [2]
 => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0}
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,1]
 => [1]
 => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,1]
 => [1]
 => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0}
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,1]
 => [1]
 => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,2]
 => [2]
 => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,1]
 => [1]
 => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [2,2,1]
 => [2,1]
 => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,2]
 => [2]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,1,1]
 => [1,1]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [4,1]
 => [1]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2]
 => [2]
 => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [4,1]
 => [1]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,1]
 => [1]
 => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [4,1]
 => [1]
 => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [4,1]
 => [1]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [2,2,1]
 => [2,1]
 => 0
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [4,1]
 => [1]
 => 0
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 0
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 0
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [4,1]
 => [1]
 => 0
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 0
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 0
[[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,.],[[.,.],[.,.]]]]
 => [2,4,6,5,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,.],[[.,[.,.]],.]]]
 => [2,5,4,6,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[.,.],.],[[.,.],.]]]
 => [2,3,5,6,4,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,[[.,.],[.,.]]],.]]
 => [3,5,4,2,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,[[.,[.,.]],.]],.]]
 => [4,3,5,2,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[.,.],[[.,.],.]],.]]
 => [2,4,5,3,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[.,[[.,.],.]],.],.]]
 => [3,4,2,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[[[.,.],.],.],.],.]]
 => [2,3,4,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [4,5,7,6,3,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [5,4,6,7,3,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => [3,5,6,7,4,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,.],.],.],[.,.]]]]
 => [3,4,5,7,6,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [3,4,6,5,7,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => [3,5,4,6,7,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => [4,3,5,6,7,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [2,6,5,7,4,3,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [2,5,4,7,6,3,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[[[[.,.],.],.],.]]]
 => [2,4,5,6,7,3,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],.],[[.,.],[.,.]]]]
 => [2,3,5,7,6,4,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],.],[[.,[.,.]],.]]]
 => [2,3,6,5,7,4,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[[.,.],.]],[[.,.],.]]]
 => [3,4,2,6,7,5,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[.,.],.],.],[[.,.],.]]]
 => [2,3,4,6,7,5,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[.,[[.,.],[.,.]]]],.]]
 => [4,6,5,3,2,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[.,[[.,[.,.]],.]]],.]]
 => [5,4,6,3,2,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[[.,[.,.]],[.,.]]],.]]
 => [4,3,6,5,2,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[[[[.,.],.],.],.]],.]]
 => [3,4,5,6,2,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => [2,4,6,5,3,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],[[.,[.,.]],.]],.]]
 => [2,5,4,6,3,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[.,.],.],[[.,.],.]],.]]
 => [2,3,5,6,4,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,[[.,.],[.,.]]],.],.]]
 => [3,5,4,2,6,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,[[.,[.,.]],.]],.],.]]
 => [4,3,5,2,6,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[.,.],[[.,.],.]],.],.]]
 => [2,4,5,3,6,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given

$\lambda$ count how many ''integer partitions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex.

Matching statistic: St000206
(load all 3 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 92%●distinct values known / distinct values provided: 50%

Values

[.,.]
 => [1] => [1]
 => []
 => ? = 0
[.,[.,.]]
 => [2,1] => [2]
 => []
 => ? = 0
[[.,.],.]
 => [1,2] => [1,1]
 => [1]
 => 0
[.,[.,[.,.]]]
 => [3,2,1] => [2,1]
 => [1]
 => 0
[.,[[.,.],.]]
 => [2,3,1] => [3]
 => []
 => ? = 0
[[.,.],[.,.]]
 => [1,3,2] => [2,1]
 => [1]
 => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1]
 => [1]
 => 0
[[[.,.],.],.]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [2,2]
 => [2]
 => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4]
 => []
 => ? ∊ {0,0}
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,1]
 => [1]
 => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,1]
 => [1]
 => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0}
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,1]
 => [1]
 => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,2]
 => [2]
 => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,1]
 => [1]
 => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [2,2,1]
 => [2,1]
 => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,2]
 => [2]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,1,1]
 => [1,1]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [4,1]
 => [1]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2]
 => [2]
 => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [4,1]
 => [1]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,1]
 => [1]
 => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,1}
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [4,1]
 => [1]
 => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [4,1]
 => [1]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [2,2,1]
 => [2,1]
 => 0
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [4,1]
 => [1]
 => 0
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 0
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 0
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [4,1]
 => [1]
 => 0
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 0
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 0
[[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,.],[[.,.],[.,.]]]]
 => [2,4,6,5,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,.],[[.,[.,.]],.]]]
 => [2,5,4,6,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[.,.],.],[[.,.],.]]]
 => [2,3,5,6,4,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,[[.,.],[.,.]]],.]]
 => [3,5,4,2,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,[[.,[.,.]],.]],.]]
 => [4,3,5,2,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[.,.],[[.,.],.]],.]]
 => [2,4,5,3,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[.,[[.,.],.]],.],.]]
 => [3,4,2,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[[[.,.],.],.],.],.]]
 => [2,3,4,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [4,7,6,5,3,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[.,[[[.,.],.],[.,.]]]]]
 => [4,5,7,6,3,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [5,4,6,7,3,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => [3,5,6,7,4,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,.],.],.],[.,.]]]]
 => [3,4,5,7,6,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [3,4,6,5,7,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => [3,5,4,6,7,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => [4,3,5,6,7,2,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [2,5,7,6,4,3,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [2,6,5,7,4,3,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [2,5,4,7,6,3,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[[[[.,.],.],.],.]]]
 => [2,4,5,6,7,3,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],.],[[.,.],[.,.]]]]
 => [2,3,5,7,6,4,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],.],[[.,[.,.]],.]]]
 => [2,3,6,5,7,4,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[[.,.],.]],[[.,.],.]]]
 => [3,4,2,6,7,5,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[.,.],.],.],[[.,.],.]]]
 => [2,3,4,6,7,5,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[.,[[.,.],[.,.]]]],.]]
 => [4,6,5,3,2,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[.,[[.,[.,.]],.]]],.]]
 => [5,4,6,3,2,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[[.,[.,.]],[.,.]]],.]]
 => [4,3,6,5,2,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[[[[.,.],.],.],.]],.]]
 => [3,4,5,6,2,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => [2,4,6,5,3,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],[[.,[.,.]],.]],.]]
 => [2,5,4,6,3,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[.,.],.],[[.,.],.]],.]]
 => [2,3,5,6,4,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,[[.,.],[.,.]]],.],.]]
 => [3,5,4,2,6,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,[[.,[.,.]],.]],.],.]]
 => [4,3,5,2,6,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[.,.],[[.,.],.]],.],.]]
 => [2,4,5,3,6,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given

$\lambda$ count how many ''integer compositions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].

Matching statistic: St001651
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001651: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 89%●distinct values known / distinct values provided: 25%

Values

[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 0
[.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? = 0
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0}
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0}
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1}
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1}
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1}
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1}
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1}
[[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1}
[.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[.,[[.,.],[.,.]]],.]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[.,[[[[.,.],[.,.]],.],.]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[.,.],[[[.,.],[.,.]],.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[.,[.,[[.,.],[.,.]]]],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[.,[[[.,.],[.,.]],.]],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[[.,.],[[.,.],[.,.]]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[[[.,.],[.,.]],[.,.]],.]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[[.,[[.,.],[.,.]]],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2}
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[.,[[[.,.],[.,.]],.]]]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[.,.],[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[.,[[[[.,.],[.,.]],.],.]]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[.,[[.,.],[.,.]]]],.]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[.,[[[.,.],[.,.]],.]],.]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[.,[[.,.],[.,.]]],.],.]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[.,[[[[[.,.],[.,.]],.],.],.]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[.,.],[.,[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[.,.],[[.,[[.,.],[.,.]]],.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[.,.],[[[[.,.],[.,.]],.],.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[[.,.],[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[[.,[.,[[.,.],[.,.]]]],[.,.]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}

Description

The Frankl number of a lattice. For a lattice

$L$ on at least two elements, this is

$\max_x(|L|-2|[x, 1]|),$ where we maximize over all join irreducible elements and

$[x, 1]$ denotes the interval from

$x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if

$L$ is a Boolean lattice.

Matching statistic: St000731

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [1,2] => [1,2] => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[[.,.],[.,.]]
 => [3,1,2] => [1,2,3] => [1,2,3] => 0
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => [1,3,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 0
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => 0
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[.,[.,.]]]],[[.,.],.]]
 => [6,7,4,3,2,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[.,.],.]]],[.,[.,.]]]
 => [7,6,3,4,2,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[.,.],.]]],[[.,.],.]]
 => [6,7,3,4,2,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,.],[.,.]]],[.,[.,.]]]
 => [7,6,4,2,3,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,.],[.,.]]],[[.,.],.]]
 => [6,7,4,2,3,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[[.,.],.],.]],[.,[.,.]]]
 => [7,6,2,3,4,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[[.,.],.],.]],[[.,.],.]]
 => [6,7,2,3,4,1,5] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,.]],[.,.]],[.,[.,.]]]
 => [7,6,4,2,1,3,5] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,.]],[.,.]],[[.,.],.]]
 => [6,7,4,2,1,3,5] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,[.,.]]],.],[.,[.,.]]]
 => [7,6,3,2,1,4,5] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,[.,.]]],.],[[.,.],.]]
 => [6,7,3,2,1,4,5] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[[.,.],.]],.],[.,[.,.]]]
 => [7,6,2,3,1,4,5] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[[.,.],.]],.],[[.,.],.]]
 => [6,7,2,3,1,4,5] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[[.,[.,.]],.],.],[.,[.,.]]]
 => [7,6,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[[.,[.,.]],.],.],[[.,.],.]]
 => [6,7,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[.,[[.,.],.]]]],[.,.]]
 => [7,4,5,3,2,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[.,.],[.,.]]]],[.,.]]
 => [7,5,3,4,2,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[.,[.,.]],.]]],[.,.]]
 => [7,4,3,5,2,1,6] => [1,6,2,3,5,4,7] => [1,5,2,3,6,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[[.,.],.],.]]],[.,.]]
 => [7,3,4,5,2,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,.],[.,[.,.]]]],[.,.]]
 => [7,5,4,2,3,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,.],[[.,.],.]]],[.,.]]
 => [7,4,5,2,3,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[[.,.],.],[.,.]]],[.,.]]
 => [7,5,2,3,4,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,[.,[.,.]]],.]],[.,.]]
 => [7,4,3,2,5,1,6] => [1,6,2,5,3,4,7] => [1,5,2,6,4,3,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,[[.,.],.]],.]],[.,.]]
 => [7,3,4,2,5,1,6] => [1,6,2,5,3,4,7] => [1,5,2,6,4,3,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[[.,.],[.,.]],.]],[.,.]]
 => [7,4,2,3,5,1,6] => [1,6,2,3,5,4,7] => [1,5,2,3,6,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[[[.,.],.],.],.]],[.,.]]
 => [7,2,3,4,5,1,6] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,.]],[.,[.,.]]],[.,.]]
 => [7,5,4,2,1,3,6] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,.]],[[.,.],.]],[.,.]]
 => [7,4,5,2,1,3,6] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,[.,.]]],[.,.]],[.,.]]
 => [7,5,3,2,1,4,6] => [1,4,6,2,3,5,7] => [1,6,4,3,2,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[[.,.],.]],[.,.]],[.,.]]
 => [7,5,2,3,1,4,6] => [1,4,6,2,3,5,7] => [1,6,4,3,2,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[[.,[.,.]],.],[.,.]],[.,.]]
 => [7,5,2,1,3,4,6] => [1,3,4,6,2,5,7] => [1,6,3,4,2,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,[.,[.,.]]]],.],[.,.]]
 => [7,4,3,2,1,5,6] => [1,5,6,2,3,4,7] => [1,6,5,3,4,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[.,[[.,.],.]]],.],[.,.]]
 => [7,3,4,2,1,5,6] => [1,5,6,2,3,4,7] => [1,6,5,3,4,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[[.,.],[.,.]]],.],[.,.]]
 => [7,4,2,3,1,5,6] => [1,5,6,2,3,4,7] => [1,6,5,3,4,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[[.,[.,.]],.]],.],[.,.]]
 => [7,3,2,4,1,5,6] => [1,5,6,2,4,3,7] => [1,6,5,3,2,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[.,[[[.,.],.],.]],.],[.,.]]
 => [7,2,3,4,1,5,6] => [1,5,6,2,3,4,7] => [1,6,5,3,4,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[[.,[.,.]],[.,.]],.],[.,.]]
 => [7,4,2,1,3,5,6] => [1,3,5,6,2,4,7] => [1,6,3,5,4,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[[.,[.,[.,.]]],.],.],[.,.]]
 => [7,3,2,1,4,5,6] => [1,4,5,6,2,3,7] => [1,6,5,4,3,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[[.,[[.,.],.]],.],.],[.,.]]
 => [7,2,3,1,4,5,6] => [1,4,5,6,2,3,7] => [1,6,5,4,3,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[[[[.,[.,.]],.],.],.],[.,.]]
 => [7,2,1,3,4,5,6] => [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[.,[.,.]],[.,.]]]],.]
 => [6,4,3,5,2,1,7] => [1,7,2,3,5,4,6] => [1,5,2,3,7,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[.,[.,[.,.]]],.]]],.]
 => [5,4,3,6,2,1,7] => [1,7,2,3,6,4,5] => [1,6,2,3,7,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[.,[[.,.],.]],.]]],.]
 => [4,5,3,6,2,1,7] => [1,7,2,3,6,4,5] => [1,6,2,3,7,5,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[.,[[[.,[.,.]],.],.]]],.]
 => [4,3,5,6,2,1,7] => [1,7,2,3,5,6,4] => [1,5,2,3,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,[.,[.,.]]],[.,.]]],.]
 => [6,4,3,2,5,1,7] => [1,7,2,5,3,4,6] => [1,5,2,7,4,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,[[.,.],.]],[.,.]]],.]
 => [6,3,4,2,5,1,7] => [1,7,2,5,3,4,6] => [1,5,2,7,4,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[[.,.],[.,.]],[.,.]]],.]
 => [6,4,2,3,5,1,7] => [1,7,2,3,5,4,6] => [1,5,2,3,7,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}
[[.,[[.,[.,[.,[.,.]]]],.]],.]
 => [5,4,3,2,6,1,7] => [1,7,2,6,3,4,5] => [1,6,2,7,4,5,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2}

Description

The number of double exceedences of a permutation. A double exceedence is an index

$\sigma(i)$ such that

$i < \sigma(i) < \sigma(\sigma(i))$ .

Matching statistic: St000379
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000379: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 80%●distinct values known / distinct values provided: 25%

Values

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

Description

The number of Hamiltonian cycles in a graph. A Hamiltonian cycle in a graph

$G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of

$G$ . Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.

Matching statistic: St000699

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000699: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 80%●distinct values known / distinct values provided: 25%

Values

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

Description

The toughness times the least common multiple of 1,...,n-1 of a non-complete graph. A graph

$G$ is

$t$ -tough if

$G$ cannot be split into

$k$ different connected components by the removal of fewer than

$tk$ vertices for all integers

$k>1$ . The toughness of

$G$ is the maximal number

$t$ such that

$G$ is

$t$ -tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero. This statistic is the toughness multiplied by the least common multiple of

$1,\dots,n-1$ , where

$n$ is the number of vertices of

$G$ .

Matching statistic: St001175

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 80%●distinct values known / distinct values provided: 25%

Values

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

Description

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

The following 233 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001281The normalized isoperimetric number of a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001537The number of cyclic crossings of a permutation. St000914The sum of the values of the Möbius function of a poset. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000455The second largest eigenvalue of a graph if it is integral. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001371The length of the longest Yamanouchi prefix of a binary word. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001890The maximum magnitude of the Möbius function of a poset. St000406The number of occurrences of the pattern 3241 in a permutation. St000929The constant term of the character polynomial of an integer partition. St000405The number of occurrences of the pattern 1324 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000359The number of occurrences of the pattern 23-1. St000220The number of occurrences of the pattern 132 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000255The number of reduced Kogan faces with the permutation as type. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001568The smallest positive integer that does not appear twice in the partition. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St001162The minimum jump of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000878The number of ones minus the number of zeros of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001271The competition number of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001570The minimal number of edges to add to make a graph Hamiltonian. St000259The diameter of a connected graph. St000260The radius 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. St001330The hat guessing number of a graph. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000741The Colin de Verdière graph invariant. St001720The minimal length of a chain of small intervals in a lattice. St001545The second Elser number of a connected graph. St000666The number of right tethers of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000297The number of leading ones in a binary word. St000655The length of the minimal rise of a Dyck path. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001130The number of two successive successions in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001715The number of non-records in a permutation. St001866The nesting alignments of a signed permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001344The neighbouring number of a permutation. St001665The number of pure excedances of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001429The number of negative entries in a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000068The number of minimal elements in a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000629The defect of a binary word. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001490The number of connected components of a skew partition. St000382The first part of an integer composition. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001871The number of triconnected components of a graph. St000096The number of spanning trees of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000287The number of connected components of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001828The Euler characteristic of a graph. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001889The size of the connectivity set of a signed permutation. St000286The number of connected components of the complement of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000366The number of double descents of a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001115The number of even descents of a permutation. St000383The last part of an integer composition. St000696The number of cycles in the breakpoint graph of a permutation. St000007The number of saliances of the permutation. St000439The position of the first down step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St000894The trace of an alternating sign matrix. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001381The fertility of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000701The protection number of a binary tree. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000905The number of different multiplicities of parts of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001765The number of connected components of the friends and strangers graph. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000822The Hadwiger number of the graph. St000445The number of rises of length 1 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001927Sparre Andersen's number of positives of a signed permutation.