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Your data matches 105 different statistics following compositions of up to 3 maps.
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Matching statistic: St001384
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 30%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 30%
Values
[3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 0 + 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[2,5,1,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[2,5,4,1,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[3,1,5,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[3,2,1,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[3,2,5,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[3,2,5,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[3,4,2,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[3,4,2,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[3,4,5,2,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[3,5,1,4,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[3,5,2,1,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[3,5,2,4,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[3,5,4,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[4,1,5,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 2
[3,2,1,6,5,4] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[3,2,6,1,5,4] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[3,2,6,5,1,4] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[3,6,2,1,5,4] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[3,6,2,5,1,4] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,2,1,6,5,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,2,6,1,5,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,2,6,5,1,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,3,1,6,5,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,3,6,1,5,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,3,6,5,1,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,6,2,1,5,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,6,2,5,1,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,6,3,1,5,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,6,3,5,1,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,2,1,6,4,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,2,6,1,4,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,2,6,4,1,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,3,1,6,4,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,3,6,1,4,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,3,6,4,1,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,6,2,1,4,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,6,2,4,1,3] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,6,3,1,4,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[5,6,3,4,1,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,3,2,7,6,5] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,3,7,2,6,5] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,3,7,6,2,5] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,7,3,2,6,5] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,7,3,6,2,5] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,3,2,7,6,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,3,7,2,6,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,3,7,6,2,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,4,2,7,6,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,4,7,2,6,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,4,7,6,2,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,7,3,2,6,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,7,3,6,2,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,7,4,2,6,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,7,4,6,2,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,3,2,7,5,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,3,7,2,5,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,3,7,5,2,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,4,2,7,5,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,4,7,2,5,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,4,7,5,2,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,7,3,2,5,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,7,3,5,2,4] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,7,4,2,5,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
[1,6,7,4,5,2,3] => [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000365
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[3,2,1] => [1,2,3] => [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[2,4,3,1] => [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => [2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1 = 0 + 1
[4,2,1,3] => [3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[4,2,3,1] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1 = 0 + 1
[4,3,1,2] => [2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 2 = 1 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1 = 0 + 1
[1,4,5,3,2] => [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,5,2,4,3] => [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1 = 0 + 1
[1,5,3,2,4] => [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,5,3,4,2] => [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1 = 0 + 1
[1,5,4,2,3] => [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 1 = 0 + 1
[1,5,4,3,2] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
[2,1,5,4,3] => [3,4,5,1,2] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1 = 0 + 1
[2,3,5,4,1] => [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[2,4,3,1,5] => [5,1,3,4,2] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 1 = 0 + 1
[2,4,3,5,1] => [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1 = 0 + 1
[2,4,5,3,1] => [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[2,5,1,4,3] => [3,4,1,5,2] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1 = 0 + 1
[2,5,3,1,4] => [4,1,3,5,2] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 1 = 0 + 1
[2,5,3,4,1] => [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1 = 0 + 1
[2,5,4,1,3] => [3,1,4,5,2] => [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,5,4,3,1] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
[3,1,5,4,2] => [2,4,5,1,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1 = 0 + 1
[3,2,1,4,5] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1 = 0 + 1
[3,2,1,5,4] => [4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1 = 0 + 1
[3,2,4,1,5] => [5,1,4,2,3] => [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 1 = 0 + 1
[3,2,4,5,1] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 1 = 0 + 1
[3,2,5,1,4] => [4,1,5,2,3] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 1 = 0 + 1
[3,2,5,4,1] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1 = 0 + 1
[3,4,2,1,5] => [5,1,2,4,3] => [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 1 = 0 + 1
[3,4,2,5,1] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1 = 0 + 1
[3,4,5,2,1] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[3,5,1,4,2] => [2,4,1,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1 = 0 + 1
[3,5,2,1,4] => [4,1,2,5,3] => [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,5,2,4,1] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1 = 0 + 1
[3,5,4,1,2] => [2,1,4,5,3] => [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,5,4,2,1] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 2 = 1 + 1
[4,1,3,2,5] => [5,2,3,1,4] => [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 1 = 0 + 1
[4,1,3,5,2] => [2,5,3,1,4] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 1 = 0 + 1
[4,1,5,3,2] => [2,3,5,1,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1 = 0 + 1
[4,2,1,3,5] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1 = 0 + 1
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ? = 0 + 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [[1,2,4],[3],[5],[6],[7]]
=> [7,6,5,3,1,2,4] => ? = 0 + 1
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [[1,2,4],[3],[5],[6],[7]]
=> [7,6,5,3,1,2,4] => ? = 0 + 1
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ? = 0 + 1
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [[1,2,4],[3],[5],[6],[7]]
=> [7,6,5,3,1,2,4] => ? = 0 + 1
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ? = 0 + 1
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1 + 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0 + 1
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ? = 0 + 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [[1,2,4],[3],[5],[6],[7]]
=> [7,6,5,3,1,2,4] => ? = 0 + 1
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [[1,2,4],[3,5],[6],[7]]
=> [7,6,3,5,1,2,4] => ? = 0 + 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ? = 0 + 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [[1,2,4],[3],[5],[6],[7]]
=> [7,6,5,3,1,2,4] => ? = 0 + 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [[1,3,4],[2,5],[6],[7]]
=> [7,6,2,5,1,3,4] => ? = 0 + 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1 + 1
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0 + 1
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => ? = 0 + 1
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [[1,2,5],[3,4],[6],[7]]
=> [7,6,3,4,1,2,5] => ? = 0 + 1
[1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => ? = 0 + 1
[1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [[1,2,5],[3],[4],[6],[7]]
=> [7,6,4,3,1,2,5] => ? = 0 + 1
[1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => ? = 0 + 1
[1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0 + 1
[1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ? = 0 + 1
[1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [[1,2,4],[3],[5],[6],[7]]
=> [7,6,5,3,1,2,4] => ? = 0 + 1
[1,2,5,6,7,4,3] => [3,4,7,6,5,2,1] => [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => [[1,2,4],[3,5],[6],[7]]
=> [7,6,3,5,1,2,4] => ? = 0 + 1
[1,2,5,7,4,3,6] => [6,3,4,7,5,2,1] => [[1,3,4],[2,5],[6],[7]]
=> [7,6,2,5,1,3,4] => ? = 0 + 1
[1,2,5,7,4,6,3] => [3,6,4,7,5,2,1] => [[1,2,4],[3,5],[6],[7]]
=> [7,6,3,5,1,2,4] => ? = 0 + 1
[1,2,5,7,6,3,4] => [4,3,6,7,5,2,1] => [[1,3,4],[2,5],[6],[7]]
=> [7,6,2,5,1,3,4] => ? = 0 + 1
[1,2,5,7,6,4,3] => [3,4,6,7,5,2,1] => [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1 + 1
[1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => [[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => ? = 0 + 1
[1,2,6,3,5,7,4] => [4,7,5,3,6,2,1] => [[1,2,5],[3],[4],[6],[7]]
=> [7,6,4,3,1,2,5] => ? = 0 + 1
[1,2,6,3,7,5,4] => [4,5,7,3,6,2,1] => [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0 + 1
[1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => [[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => ? = 0 + 1
[1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => [[1,2,5],[3,4],[6],[7]]
=> [7,6,3,4,1,2,5] => ? = 0 + 1
[1,2,6,4,5,3,7] => [7,3,5,4,6,2,1] => [[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => ? = 0 + 1
[1,2,6,4,5,7,3] => [3,7,5,4,6,2,1] => [[1,2,5],[3],[4],[6],[7]]
=> [7,6,4,3,1,2,5] => ? = 0 + 1
[1,2,6,4,7,3,5] => [5,3,7,4,6,2,1] => [[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => ? = 0 + 1
[1,2,6,4,7,5,3] => [3,5,7,4,6,2,1] => [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0 + 1
[1,2,6,5,3,4,7] => [7,4,3,5,6,2,1] => [[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => ? = 0 + 1
[1,2,6,5,3,7,4] => [4,7,3,5,6,2,1] => [[1,2,5],[3,4],[6],[7]]
=> [7,6,3,4,1,2,5] => ? = 0 + 1
[1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => ? = 1 + 1
[1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => [[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => ? = 1 + 1
[1,2,6,5,7,3,4] => [4,3,7,5,6,2,1] => [[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => ? = 0 + 1
[1,2,6,5,7,4,3] => [3,4,7,5,6,2,1] => [[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ? = 1 + 1
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St000864
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 40%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 40%
Values
[3,2,1] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 0 + 2
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[2,4,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[3,2,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[3,4,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[4,1,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[4,2,1,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[4,3,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 0 + 2
[4,3,2,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 1 + 2
[1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,4,5,3,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,5,2,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,5,3,2,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,5,3,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,5,4,2,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,5,4,3,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 1 + 2
[2,1,5,4,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[2,3,5,4,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[2,4,3,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[2,4,3,5,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[2,4,5,3,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[2,5,1,4,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[2,5,3,1,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[2,5,3,4,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[2,5,4,1,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[2,5,4,3,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 1 + 2
[3,1,5,4,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[3,2,1,4,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[3,2,1,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[3,2,4,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[3,2,4,5,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[3,2,5,1,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[3,2,5,4,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[3,4,2,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[3,4,2,5,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[3,4,5,2,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[3,5,1,4,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[3,5,2,1,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[3,5,2,4,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[3,5,4,1,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[3,5,4,2,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 1 + 2
[4,1,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[4,1,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[4,1,5,3,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 0 + 2
[4,2,1,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,2,3,4,7,6,5] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,5,7,6,4] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,6,5,4,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,6,5,7,4] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,6,7,5,4] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,7,4,6,5] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,7,5,4,6] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,7,5,6,4] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,7,6,4,5] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,3,7,6,5,4] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 2
[1,2,4,3,7,6,5] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,4,5,7,6,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,4,6,5,3,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,4,6,5,7,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,4,6,7,5,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,4,7,3,6,5] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,4,7,5,3,6] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,4,7,5,6,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,4,7,6,3,5] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,4,7,6,5,3] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 2
[1,2,5,3,7,6,4] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,5,4,3,6,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,5,4,3,7,6] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,5,4,6,3,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,5,4,6,7,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,5,4,7,3,6] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,5,4,7,6,3] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,5,6,4,3,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,5,6,4,7,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,5,6,7,4,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,5,7,3,6,4] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,5,7,4,3,6] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,5,7,4,6,3] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,5,7,6,3,4] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,5,7,6,4,3] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 2
[1,2,6,3,5,4,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,6,3,5,7,4] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,6,3,7,5,4] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,6,4,3,5,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,6,4,3,7,5] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,6,4,5,3,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,6,4,5,7,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,6,4,7,3,5] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,6,4,7,5,3] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,6,5,3,4,7] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0 + 2
[1,2,6,5,3,7,4] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,6,5,4,3,7] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 2
[1,2,6,5,4,7,3] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 2
[1,2,6,5,7,3,4] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[1,2,6,5,7,4,3] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 + 2
Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.
Matching statistic: St001336
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 0 + 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
[1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,5,7,6,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,6,5,7,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,6,7,5,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,7,4,6,5] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,7,5,4,6] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,7,5,6,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,7,6,4,5] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,3,7,6,5,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,2,4,3,7,6,5] => ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,5,7,6,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,6,5,3,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,6,5,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,6,7,5,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,7,5,3,6] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,7,5,6,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,7,6,3,5] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,4,7,6,5,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,2,5,3,7,6,4] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7)
=> ([(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,4,3,7,6] => ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,4,6,3,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,4,6,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,4,7,3,6] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,4,7,6,3] => ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,6,4,3,7] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,6,4,7,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,6,7,4,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,7,3,6,4] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[1,2,5,7,4,3,6] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,7,4,6,3] => ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,5,7,6,3,4] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 0 + 1
[1,2,5,7,6,4,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,2,6,3,5,4,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,3,5,7,4] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,3,7,5,4] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,4,3,5,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,4,3,7,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,4,5,3,7] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,4,5,7,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,4,7,3,5] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[1,2,6,4,7,5,3] => ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,5,3,4,7] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,5,3,7,4] => ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,2,6,5,4,3,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,2,6,5,4,7,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,2,6,5,7,3,4] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 0 + 1
Description
The minimal number of vertices in a graph whose complement is triangle-free.
Matching statistic: St001330
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 60%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 60%
Values
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[3,5,2,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,5,4,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
[4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,2,3,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
[4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
[4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,5,2,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,5,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
[5,1,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[5,1,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[5,1,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[5,1,4,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[5,1,4,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
[5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[5,2,1,4,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[5,2,3,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[5,2,3,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[5,2,4,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[5,2,4,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
[5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[5,3,1,4,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[5,3,2,1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[5,3,2,4,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
[5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 3
[5,3,4,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
[5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[5,4,1,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
[5,4,2,1,3] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[5,4,2,3,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 3
[5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[5,4,3,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 2 + 3
[1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,2,4,6,5,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,2,5,4,3,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,2,5,4,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 3
[1,2,5,6,4,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,2,6,4,3,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,2,6,5,3,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,2,6,5,4,3] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 1 + 3
[1,3,4,6,5,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,3,5,4,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,3,5,6,4,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,5,2,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,5,4,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,2,3,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[4,5,2,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,5,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,1,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[5,1,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[5,1,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[5,1,4,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[5,1,4,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[5,2,1,4,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[5,2,3,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[5,2,3,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[5,2,4,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[5,2,4,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[5,3,1,4,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[5,3,2,1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[5,3,2,4,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[5,3,4,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[5,4,1,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,4,2,1,3] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[5,4,2,3,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[5,4,3,2,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,4,6,5,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,5,4,3,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,5,4,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,2,5,6,4,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,6,4,3,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,6,5,3,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,6,5,4,3] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,3,4,6,5,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,3,5,4,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,5,6,4,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000508
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000508: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000508: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Values
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,5,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[3,1,5,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[4,2,1,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[4,2,5,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,3,2,1,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,3,2,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,3,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 2
[1,2,3,6,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,4,6,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,5,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,5,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,5,6,4,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,6,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,6,4,3,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,6,4,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,6,5,3,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
[1,3,4,6,5,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,3,5,4,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,3,5,4,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,3,5,6,4,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,3,6,4,2,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,3,6,4,5,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
[1,4,3,2,5,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,4,3,5,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,4,3,5,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,4,5,3,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,4,5,3,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,4,5,6,3,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,4,6,5,3,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
[1,5,2,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,5,2,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,5,3,2,4,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,5,3,4,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,5,3,4,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,5,4,2,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
[1,5,4,3,6,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
Description
Eigenvalues of the random-to-random operator acting on a simple module.
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module [1].
Matching statistic: St001001
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Values
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,5,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[3,1,5,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,2,1,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,2,5,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,3,2,1,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,3,2,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,3,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,2,3,6,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,4,6,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,5,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,5,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,5,6,4,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,6,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,6,4,3,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,6,4,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,6,5,3,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
[1,3,4,6,5,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,3,5,4,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,3,5,4,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,3,5,6,4,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,3,6,4,2,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,3,6,4,5,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
[1,4,3,2,5,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,4,3,5,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,4,3,5,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,4,5,3,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,4,5,3,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,4,5,6,3,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,4,6,5,3,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
[1,5,2,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,5,2,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,5,3,2,4,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,5,3,4,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,5,3,4,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,5,4,2,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
[1,5,4,3,6,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Values
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 1
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,3,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,4,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,4,5,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,5,1,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,5,3,1,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,5,4,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[3,1,5,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,2,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,2,4,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,2,5,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,2,5,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,4,2,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,4,2,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,4,5,2,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,5,1,4,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,5,2,1,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,5,2,4,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,5,4,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[4,1,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,1,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,1,5,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[4,2,1,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,2,1,5,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[4,2,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,2,3,5,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[4,2,5,1,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,3,2,1,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,3,2,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,3,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 2
[1,2,3,6,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,4,6,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,5,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,5,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,5,6,4,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,6,3,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,6,4,3,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,6,4,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,6,5,3,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,2,6,5,4,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
[1,3,4,6,5,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,3,5,4,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,3,5,4,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,3,5,6,4,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,3,6,4,2,5] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,3,6,4,5,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,3,6,5,4,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
[1,4,3,2,5,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,4,3,5,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,4,3,5,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,4,5,3,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,4,5,3,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,4,5,6,3,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,4,6,5,3,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
[1,5,2,4,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,5,2,4,6,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,5,3,2,4,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,5,3,4,2,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,5,3,4,6,2] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,5,4,2,3,6] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[1,5,4,3,2,6] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
[1,5,4,3,6,2] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
The following 95 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001557The number of inversions of the second entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001651The Frankl number of a lattice. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001561The value of the elementary symmetric function evaluated at 1. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St000475The number of parts equal to 1 in a partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000480The number of lower covers of a partition in dominance order. St000759The smallest missing part in an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000481The number of upper covers of a partition in dominance order. St001568The smallest positive integer that does not appear twice in the partition. St001845The number of join irreducibles minus the rank of a lattice. St001644The dimension of a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000256The number of parts from which one can substract 2 and still get an integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000177The number of free tiles in the pattern. St000178Number of free entries. St001960The number of descents of a permutation minus one if its first entry is not one. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000741The Colin de Verdière graph invariant. St001626The number of maximal proper sublattices of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. 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. St001618The cardinality of the Frattini sublattice of a lattice. St001875The number of simple modules with projective dimension at most 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000699The toughness times the least common multiple of 1,. St001625The Möbius invariant of a lattice. St001964The interval resolution global dimension of a poset. 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. St001896The number of right descents of a signed permutations. St001615The number of join prime elements of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001754The number of tolerances of a finite lattice. St001622The number of join-irreducible elements of a lattice. St001623The number of doubly irreducible elements of a lattice. 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. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001833The number of linear intervals in a lattice. St001620The number of sublattices of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St000068The number of minimal elements in a poset. St001866The nesting alignments of a signed permutation. St001864The number of excedances of a signed permutation. St000069The number of maximal elements of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001851The number of Hecke atoms of a signed permutation. St001895The oddness of a signed permutation. St001862The number of crossings of a signed permutation. St001769The reflection length of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition.
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