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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St001932
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition.
Let $D$ be a Dyck path, and let $P$ be the noncrossing set partition obtained by applying [[Mp00138]]. For each pair of singleton blocks $\{a\}, \{b\}$, let $P'$ be the set partition obtained from $P$ by merging the two blocks. This statistic enumerates the number of (unordered) pairs of singleton blocks such that $P'$ is noncrossing.
Matching statistic: St000790
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000790: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 88%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000790: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 88%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 15
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 15
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 15
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 10
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? = 15
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 6
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 10
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> ? = 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 10
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
Description
The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path.
Apparently, the total number of these is given in [1].
The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St000456
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 19%●distinct values known / distinct values provided: 12%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 19%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([],2)
=> ? = 1 + 1
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([],3)
=> ? = 3 + 1
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> ? = 6 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> ? = 10 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([],6)
=> ([],6)
=> ? = 15 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? = 3 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001545
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 13%●distinct values known / distinct values provided: 12%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 13%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 0 + 2
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],1)
=> ? = 1 + 2
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> ? = 3 + 2
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 6 + 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 10 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 15 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? = 6 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St000804
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000804: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 75%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000804: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 6
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => 10
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> [3,1,2,4,5,6] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[[[.,.],[[.,.],.]],.],.]
=> [3,4,1,2,5,6] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> [4,2,1,3,5,6] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> [4,1,2,3,5,6] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[[[.,.],.],.]],.]
=> [3,4,5,1,2,6] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> [5,3,4,1,2,6] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],[[.,.],.]],.]
=> [4,5,2,1,3,6] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> [5,3,2,1,4,6] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> [5,2,3,1,4,6] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[[.,.],.]],.]
=> [4,5,1,2,3,6] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> [5,2,1,3,4,6] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> [5,1,2,3,4,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> [3,4,5,6,1,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,.],[.,.]],.]]
=> [5,3,4,6,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],[[.,.],.]]]
=> [5,6,3,4,1,2] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> [6,4,3,5,1,2] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [6,3,4,5,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> [4,5,6,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> [6,4,5,2,1,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> [5,6,3,2,1,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [6,4,3,2,1,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> [6,3,4,2,1,5] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[.,[[.,.],.]],[[.,.],.]]
=> [5,6,2,3,1,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> [6,3,2,4,1,5] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> [6,2,3,4,1,5] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> [6,4,2,3,1,5] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[[[[.,.],.],[.,.]],.],.],.]
=> [4,1,2,3,5,6,7] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[[[.,.],[[[.,.],.],.]],.],.]
=> [3,4,5,1,2,6,7] => ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[[[.,.],[[.,.],[.,.]]],.],.]
=> [5,3,4,1,2,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[[[.,[.,.]],[[.,.],.]],.],.]
=> [4,5,2,1,3,6,7] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [5,3,2,1,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[[[.,[[.,.],.]],[.,.]],.],.]
=> [5,2,3,1,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> [4,5,1,2,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[[[[[.,.],.],.],[.,.]],.],.]
=> [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[[.,.],[[[[.,.],.],.],.]],.]
=> [3,4,5,6,1,2,7] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[[.,.],[[[.,.],[.,.]],.]],.]
=> [5,3,4,6,1,2,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],[[.,.],.]]],.]
=> [5,6,3,4,1,2,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [[[.,.],[[.,[.,.]],[.,.]]],.]
=> [6,4,3,5,1,2,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[[.,.],[[[.,.],.],[.,.]]],.]
=> [6,3,4,5,1,2,7] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[[.,[.,.]],[[[.,.],.],.]],.]
=> [4,5,6,2,1,3,7] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [6,4,5,2,1,3,7] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[[.,[.,[.,.]]],[[.,.],.]],.]
=> [5,6,3,2,1,4,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [6,4,3,2,1,5,7] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[[.,[.,[[.,.],.]]],[.,.]],.]
=> [6,3,4,2,1,5,7] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [[[.,[[.,.],.]],[[.,.],.]],.]
=> [5,6,2,3,1,4,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [[[.,[[.,[.,.]],.]],[.,.]],.]
=> [6,3,2,4,1,5,7] => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [[[.,[[[.,.],.],.]],[.,.]],.]
=> [6,2,3,4,1,5,7] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[[.,[[.,.],[.,.]]],[.,.]],.]
=> [6,4,2,3,1,5,7] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[[[.,.],.],[[.,.],[.,.]]],.]
=> [6,4,5,1,2,3,7] => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [[[[.,[.,.]],.],[[.,.],.]],.]
=> [5,6,2,1,3,4,7] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [[[[.,[[.,.],.]],.],[.,.]],.]
=> [6,2,3,1,4,5,7] => ? = 0
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> [6,1,2,3,4,5,7] => ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [[[[.,.],[[.,.],.]],[.,.]],.]
=> [6,3,4,1,2,5,7] => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 0
[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]
=> [[[[.,.],[.,.]],[[.,.],.]],.]
=> [5,6,3,1,2,4,7] => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => ? = 6
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [[.,.],[[[[.,.],[.,.]],.],.]]
=> [5,3,4,6,7,1,2] => ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [[.,.],[[[.,.],[[.,.],.]],.]]
=> [5,6,3,4,7,1,2] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [[.,.],[[[.,[.,.]],[.,.]],.]]
=> [6,4,3,5,7,1,2] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],[.,.]],.]]
=> [6,3,4,5,7,1,2] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[.,.],[[[.,.],.],.]]]
=> [5,6,7,3,4,1,2] => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => ? = 0
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 0
Description
The number of occurrences of the vincular pattern |123 in a permutation.
This is the number of occurrences of the pattern $(1,2,3)$, such that the letter matched by $1$ is the first entry of the permutation.
Matching statistic: St001198
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 0 + 2
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 1 + 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 3 + 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 10 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 15 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 0 + 2
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 1 + 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 3 + 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 10 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 15 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000264
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 0 + 4
[1,0,1,0]
=> [1,2] => ([],2)
=> ([],1)
=> ? = 1 + 4
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 4
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> ? = 3 + 4
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 4
[1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 6 + 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 4
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 4
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 4
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 10 + 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 0 + 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 4
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,2,4,6] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,2,7,3,5] => ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,7,4] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,1,3,7,5] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,5,1,6,3,7,4] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,1,5,7,2,4,6] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,1,7,2,5] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [3,5,1,6,2,4,7] => ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [3,5,1,6,2,7,4] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,1,6,7,2,4] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [3,5,1,7,2,4,6] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,1,7,2,4] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,1,2,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,1,7,2,4,5] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [4,1,5,2,6,7,3] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,7,3,6] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [4,1,5,6,2,7,3] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,1,2,7,3,5] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,1,7,2,3,5] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,1,2,6,3,7,4] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,1,6,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,1,6,2,7,3,4] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
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