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Your data matches 24 different statistics following compositions of up to 3 maps.
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Matching statistic: St000711
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St000647
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [3,2,1] => 0
[3,1,2] => [2,3,1] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 0
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 1
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 2
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 2
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 1
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 0
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 2
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,3,4,2] => 2
[1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 0
[1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => [1,5,4,3,2,7,6] => ? = 0
[1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 0
[1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => [1,7,5,4,3,2,6] => ? = 1
[1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => [1,6,4,3,2,5,7] => ? = 1
[1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => [1,7,6,4,3,2,5] => ? = 1
[1,3,4,6,5,2,7] => [1,6,2,3,5,4,7] => [1,5,6,4,3,2,7] => ? = 1
[1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => [1,6,4,3,2,7,5] => ? = 2
[1,3,4,6,7,5,2] => [1,7,2,3,6,4,5] => [1,7,5,6,4,3,2] => ? = 2
[1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 1
[1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => [1,6,7,4,3,2,5] => ? = 1
[1,3,4,7,5,2,6] => [1,6,2,3,5,7,4] => [1,5,7,4,3,2,6] => ? = 1
[1,3,4,7,6,2,5] => [1,6,2,3,7,5,4] => [1,7,4,3,2,6,5] => ? = 1
[1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [1,5,3,2,4,6,7] => ? = 1
[1,3,5,2,4,7,6] => [1,4,2,5,3,7,6] => [1,5,3,2,4,7,6] => ? = 1
[1,3,5,2,6,4,7] => [1,4,2,6,3,5,7] => [1,6,5,3,2,4,7] => ? = 1
[1,3,5,2,6,7,4] => [1,4,2,7,3,5,6] => [1,7,6,5,3,2,4] => ? = 1
[1,3,5,2,7,4,6] => [1,4,2,6,3,7,5] => [1,7,5,3,2,4,6] => ? = 2
[1,3,5,2,7,6,4] => [1,4,2,7,3,6,5] => [1,6,7,5,3,2,4] => ? = 2
[1,3,5,4,7,2,6] => [1,6,2,4,3,7,5] => [1,4,7,5,3,2,6] => ? = 2
[1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => [1,5,3,2,6,4,7] => ? = 2
[1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => [1,5,3,2,7,6,4] => ? = 2
[1,3,5,6,4,2,7] => [1,6,2,5,3,4,7] => [1,6,4,5,3,2,7] => ? = 2
[1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => [1,7,5,3,2,6,4] => ? = 3
[1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => [1,6,4,7,5,3,2] => ? = 3
[1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => [1,5,3,2,7,4,6] => ? = 2
[1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => [1,5,3,2,6,7,4] => ? = 2
[1,3,5,7,4,2,6] => [1,6,2,5,3,7,4] => [1,7,4,5,3,2,6] => ? = 2
[1,3,5,7,4,6,2] => [1,7,2,5,3,6,4] => [1,6,7,4,5,3,2] => ? = 2
[1,3,5,7,6,2,4] => [1,6,2,7,3,5,4] => [1,6,5,3,2,7,4] => ? = 2
[1,3,6,2,4,5,7] => [1,4,2,5,6,3,7] => [1,6,3,2,4,5,7] => ? = 1
[1,3,6,2,4,7,5] => [1,4,2,5,7,3,6] => [1,7,6,3,2,4,5] => ? = 1
[1,3,6,2,5,4,7] => [1,4,2,6,5,3,7] => [1,5,6,3,2,4,7] => ? = 1
[1,3,6,2,5,7,4] => [1,4,2,7,5,3,6] => [1,5,7,6,3,2,4] => ? = 1
[1,3,6,2,7,4,5] => [1,4,2,6,7,3,5] => [1,6,3,2,4,7,5] => ? = 2
[1,3,6,2,7,5,4] => [1,4,2,7,6,3,5] => [1,7,5,6,3,2,4] => ? = 2
[1,3,6,4,2,7,5] => [1,5,2,4,7,3,6] => [1,4,7,6,3,2,5] => ? = 1
[1,3,6,4,7,5,2] => [1,7,2,4,6,3,5] => [1,4,7,5,6,3,2] => ? = 2
[1,3,6,5,2,4,7] => [1,5,2,6,4,3,7] => [1,6,3,2,5,4,7] => ? = 1
[1,3,6,5,2,7,4] => [1,5,2,7,4,3,6] => [1,7,6,3,2,5,4] => ? = 1
[1,3,6,5,4,2,7] => [1,6,2,5,4,3,7] => [1,5,4,6,3,2,7] => ? = 1
[1,3,6,5,7,2,4] => [1,6,2,7,4,3,5] => [1,6,3,2,7,5,4] => ? = 2
[1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => [1,7,4,6,3,2,5] => ? = 2
[1,3,6,7,2,5,4] => [1,5,2,7,6,3,4] => [1,6,3,2,5,7,4] => ? = 2
[1,3,6,7,4,2,5] => [1,6,2,5,7,3,4] => [1,6,3,2,7,4,5] => ? = 2
[1,3,6,7,5,2,4] => [1,6,2,7,5,3,4] => [1,5,6,3,2,7,4] => ? = 2
[1,3,6,7,5,4,2] => [1,7,2,6,5,3,4] => [1,5,7,4,6,3,2] => ? = 2
[1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[1,3,7,2,4,6,5] => [1,4,2,5,7,6,3] => [1,6,7,3,2,4,5] => ? = 1
[1,3,7,2,5,4,6] => [1,4,2,6,5,7,3] => [1,5,7,3,2,4,6] => ? = 1
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St000710
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
St000710: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => 0
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,2,3] => 0
[1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,1,2,4] => 0
[2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [4,1,3,2] => 1
[3,1,2,4] => [2,3,1,4] => 1
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [4,2,1,3] => 1
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [4,3,1,2] => 2
[4,1,2,3] => [2,3,4,1] => 1
[4,1,3,2] => [2,4,3,1] => 1
[4,2,1,3] => [3,2,4,1] => 1
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [3,4,2,1] => 1
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,5,2,4,3] => 1
[1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,5,3,2,4] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,5,3,2] => [1,5,4,2,3] => 2
[1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 0
[1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => ? = 0
[1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 0
[1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 0
[1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => ? = 1
[1,3,4,5,7,6,2] => [1,7,2,3,4,6,5] => ? = 1
[1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 1
[1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => ? = 1
[1,3,4,6,5,2,7] => [1,6,2,3,5,4,7] => ? = 1
[1,3,4,6,5,7,2] => [1,7,2,3,5,4,6] => ? = 1
[1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => ? = 2
[1,3,4,6,7,5,2] => [1,7,2,3,6,4,5] => ? = 2
[1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 1
[1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => ? = 1
[1,3,4,7,5,2,6] => [1,6,2,3,5,7,4] => ? = 1
[1,3,4,7,5,6,2] => [1,7,2,3,5,6,4] => ? = 1
[1,3,4,7,6,2,5] => [1,6,2,3,7,5,4] => ? = 1
[1,3,4,7,6,5,2] => [1,7,2,3,6,5,4] => ? = 1
[1,3,5,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 1
[1,3,5,4,2,7,6] => [1,5,2,4,3,7,6] => ? = 1
[1,3,5,4,6,2,7] => [1,6,2,4,3,5,7] => ? = 1
[1,3,5,4,6,7,2] => [1,7,2,4,3,5,6] => ? = 1
[1,3,5,4,7,2,6] => [1,6,2,4,3,7,5] => ? = 2
[1,3,5,4,7,6,2] => [1,7,2,4,3,6,5] => ? = 2
[1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 2
[1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => ? = 2
[1,3,5,6,4,2,7] => [1,6,2,5,3,4,7] => ? = 2
[1,3,5,6,4,7,2] => [1,7,2,5,3,4,6] => ? = 2
[1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 3
[1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => ? = 3
[1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 2
[1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => ? = 2
[1,3,5,7,4,2,6] => [1,6,2,5,3,7,4] => ? = 2
[1,3,5,7,4,6,2] => [1,7,2,5,3,6,4] => ? = 2
[1,3,5,7,6,2,4] => [1,6,2,7,3,5,4] => ? = 2
[1,3,5,7,6,4,2] => [1,7,2,6,3,5,4] => ? = 2
[1,3,6,4,2,5,7] => [1,5,2,4,6,3,7] => ? = 1
[1,3,6,4,2,7,5] => [1,5,2,4,7,3,6] => ? = 1
[1,3,6,4,5,2,7] => [1,6,2,4,5,3,7] => ? = 1
[1,3,6,4,5,7,2] => [1,7,2,4,5,3,6] => ? = 1
[1,3,6,4,7,2,5] => [1,6,2,4,7,3,5] => ? = 2
[1,3,6,4,7,5,2] => [1,7,2,4,6,3,5] => ? = 2
[1,3,6,5,2,4,7] => [1,5,2,6,4,3,7] => ? = 1
[1,3,6,5,2,7,4] => [1,5,2,7,4,3,6] => ? = 1
[1,3,6,5,4,2,7] => [1,6,2,5,4,3,7] => ? = 1
[1,3,6,5,4,7,2] => [1,7,2,5,4,3,6] => ? = 1
[1,3,6,5,7,2,4] => [1,6,2,7,4,3,5] => ? = 2
[1,3,6,5,7,4,2] => [1,7,2,6,4,3,5] => ? = 2
[1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 2
[1,3,6,7,2,5,4] => [1,5,2,7,6,3,4] => ? = 2
Description
The number of big deficiencies of a permutation.
A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$.
This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
Matching statistic: St000646
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[2,3,1] => [3,1,2] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [2,3,1] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [2,3,1] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 0
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 0
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => [2,1,3,4] => 1
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 1
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => [3,1,2,4] => 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 2
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 1
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 1
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => [5,1,3,4,2] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => [5,2,1,3,4] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => [5,1,2,4,3] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => [5,3,1,2,4] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => 2
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,3,4,2] => [5,1,3,2,4] => 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 0
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 0
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 0
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 0
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 0
[1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1
[1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 1
[1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 1
[1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 0
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 0
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 0
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 0
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 0
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 0
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => ? = 0
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => ? = 0
[1,2,4,5,7,3,6] => [1,2,6,3,4,7,5] => [1,2,7,5,4,3,6] => [7,6,1,3,4,5,2] => ? = 1
[1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => [1,2,6,7,5,4,3] => [7,6,2,1,3,4,5] => ? = 1
[1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,2,6,4,3,5,7] => [7,6,2,4,5,3,1] => ? = 1
[1,2,4,6,3,7,5] => [1,2,5,3,7,4,6] => [1,2,7,6,4,3,5] => [7,6,1,2,4,5,3] => ? = 1
[1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => [1,2,5,6,4,3,7] => [7,6,3,2,4,5,1] => ? = 1
[1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => [1,2,5,7,6,4,3] => [7,6,3,1,2,4,5] => ? = 1
[1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => [1,2,6,4,3,7,5] => [7,6,2,4,5,1,3] => ? = 2
[1,2,4,6,7,5,3] => [1,2,7,3,6,4,5] => [1,2,7,5,6,4,3] => [7,6,1,3,2,4,5] => ? = 2
[1,2,4,7,3,5,6] => [1,2,5,3,6,7,4] => [1,2,7,4,3,5,6] => [7,6,1,4,5,3,2] => ? = 1
[1,2,4,7,3,6,5] => [1,2,5,3,7,6,4] => [1,2,6,7,4,3,5] => [7,6,2,1,4,5,3] => ? = 1
[1,2,4,7,5,3,6] => [1,2,6,3,5,7,4] => [1,2,5,7,4,3,6] => [7,6,3,1,4,5,2] => ? = 1
[1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => [1,2,5,6,7,4,3] => [7,6,3,2,1,4,5] => ? = 1
[1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [1,2,7,4,3,6,5] => [7,6,1,4,5,2,3] => ? = 1
[1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => [1,2,6,5,7,4,3] => [7,6,2,3,1,4,5] => ? = 1
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 1
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 1
Description
The number of big ascents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$.
For the number of small ascents, see [[St000441]].
Matching statistic: St001960
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Values
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [2,3,1] => [3,1,2] => 0
[1,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [3,2,1] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 0
[1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 0
[1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 1
[1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
[2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 0
[2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 0
[2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [3,4,2,1] => [4,1,3,2] => 1
[3,1,4,2] => [3,1,2,4] => [3,2,1,4] => 1
[3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1
[3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 2
[3,4,2,1] => [1,4,2,3] => [1,4,3,2] => 2
[4,1,2,3] => [3,4,1,2] => [3,1,4,2] => 1
[4,1,3,2] => [3,1,4,2] => [4,2,1,3] => 1
[4,2,1,3] => [4,3,1,2] => [4,2,3,1] => 1
[4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 1
[4,3,1,2] => [4,1,3,2] => [3,4,2,1] => 1
[4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,2,4,3,5] => [2,3,5,4,1] => [4,5,1,2,3] => 0
[1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,2,5,3,4] => [2,3,5,1,4] => [5,4,1,2,3] => 1
[1,2,5,4,3] => [2,3,1,5,4] => [3,1,2,5,4] => 1
[1,3,2,4,5] => [2,4,3,5,1] => [3,5,1,2,4] => 0
[1,3,2,5,4] => [2,4,3,1,5] => [3,4,1,2,5] => 0
[1,3,4,2,5] => [2,5,3,4,1] => [3,4,5,1,2] => 0
[1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,3,5,2,4] => [2,5,3,1,4] => [3,5,4,1,2] => 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,4,2,3,5] => [2,4,5,3,1] => [5,1,2,4,3] => 1
[1,4,2,5,3] => [2,4,1,3,5] => [4,3,1,2,5] => 1
[1,4,3,2,5] => [2,5,4,3,1] => [4,3,5,1,2] => 1
[1,4,3,5,2] => [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,4,5,2,3] => [2,5,1,3,4] => [5,4,3,1,2] => 2
[1,4,5,3,2] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => ? = 0
[1,2,3,4,6,5] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => ? = 0
[1,2,3,5,4,6] => [2,3,4,6,5,1] => [5,6,1,2,3,4] => ? = 0
[1,2,3,5,6,4] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => ? = 0
[1,2,3,6,4,5] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => ? = 1
[1,2,3,6,5,4] => [2,3,4,1,6,5] => [4,1,2,3,6,5] => ? = 1
[1,2,4,3,5,6] => [2,3,5,4,6,1] => [4,6,1,2,3,5] => ? = 0
[1,2,4,3,6,5] => [2,3,5,4,1,6] => [4,5,1,2,3,6] => ? = 0
[1,2,4,5,3,6] => [2,3,6,4,5,1] => [4,5,6,1,2,3] => ? = 0
[1,2,4,5,6,3] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => ? = 0
[1,2,4,6,3,5] => [2,3,6,4,1,5] => [4,6,5,1,2,3] => ? = 1
[1,2,4,6,5,3] => [2,3,1,4,6,5] => [3,1,2,4,6,5] => ? = 1
[1,2,5,3,4,6] => [2,3,5,6,4,1] => [6,1,2,3,5,4] => ? = 1
[1,2,5,3,6,4] => [2,3,5,1,4,6] => [5,4,1,2,3,6] => ? = 1
[1,2,5,4,3,6] => [2,3,6,5,4,1] => [5,4,6,1,2,3] => ? = 1
[1,2,5,4,6,3] => [2,3,1,5,4,6] => [3,1,2,5,4,6] => ? = 1
[1,2,5,6,3,4] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => ? = 2
[1,2,5,6,4,3] => [2,3,1,6,4,5] => [3,1,2,6,5,4] => ? = 2
[1,2,6,3,4,5] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => ? = 1
[1,2,6,3,5,4] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => ? = 1
[1,2,6,4,3,5] => [2,3,6,5,1,4] => [6,4,5,1,2,3] => ? = 1
[1,2,6,4,5,3] => [2,3,1,5,6,4] => [3,1,2,6,4,5] => ? = 1
[1,2,6,5,3,4] => [2,3,6,1,5,4] => [5,6,4,1,2,3] => ? = 1
[1,2,6,5,4,3] => [2,3,1,6,5,4] => [3,1,2,5,6,4] => ? = 1
[1,3,2,4,5,6] => [2,4,3,5,6,1] => [3,6,1,2,4,5] => ? = 0
[1,3,2,4,6,5] => [2,4,3,5,1,6] => [3,5,1,2,4,6] => ? = 0
[1,3,2,5,4,6] => [2,4,3,6,5,1] => [3,5,6,1,2,4] => ? = 0
[1,3,2,5,6,4] => [2,4,3,1,5,6] => [3,4,1,2,5,6] => ? = 0
[1,3,2,6,4,5] => [2,4,3,6,1,5] => [3,6,5,1,2,4] => ? = 1
[1,3,2,6,5,4] => [2,4,3,1,6,5] => [3,4,1,2,6,5] => ? = 1
[1,3,4,2,5,6] => [2,5,3,4,6,1] => [3,4,6,1,2,5] => ? = 0
[1,3,4,2,6,5] => [2,5,3,4,1,6] => [3,4,5,1,2,6] => ? = 0
[1,3,4,5,2,6] => [2,6,3,4,5,1] => [3,4,5,6,1,2] => ? = 0
[1,3,4,5,6,2] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 0
[1,3,4,6,2,5] => [2,6,3,4,1,5] => [3,4,6,5,1,2] => ? = 1
[1,3,4,6,5,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => ? = 1
[1,3,5,2,4,6] => [2,5,3,6,4,1] => [3,6,1,2,5,4] => ? = 1
[1,3,5,2,6,4] => [2,5,3,1,4,6] => [3,5,4,1,2,6] => ? = 1
[1,3,5,4,2,6] => [2,6,3,5,4,1] => [3,5,4,6,1,2] => ? = 1
[1,3,5,4,6,2] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ? = 1
[1,3,5,6,2,4] => [2,6,3,1,4,5] => [3,6,5,4,1,2] => ? = 2
[1,3,5,6,4,2] => [2,1,3,6,4,5] => [2,1,3,6,5,4] => ? = 2
[1,3,6,2,4,5] => [2,5,3,6,1,4] => [3,5,1,2,6,4] => ? = 1
[1,3,6,2,5,4] => [2,5,3,1,6,4] => [3,6,4,1,2,5] => ? = 1
[1,3,6,4,2,5] => [2,6,3,5,1,4] => [3,6,4,5,1,2] => ? = 1
[1,3,6,4,5,2] => [2,1,3,5,6,4] => [2,1,3,6,4,5] => ? = 1
[1,3,6,5,2,4] => [2,6,3,1,5,4] => [3,5,6,4,1,2] => ? = 1
[1,3,6,5,4,2] => [2,1,3,6,5,4] => [2,1,3,5,6,4] => ? = 1
[1,4,2,3,5,6] => [2,4,5,3,6,1] => [6,1,2,4,3,5] => ? = 1
[1,4,2,3,6,5] => [2,4,5,3,1,6] => [5,1,2,4,3,6] => ? = 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001907
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00162: Signed permutations —inverse⟶ Signed permutations
St001907: Signed permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00162: Signed permutations —inverse⟶ Signed permutations
St001907: Signed permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,-1] => [-2,1] => 1 = 0 + 1
[2,1] => [2,1] => [1,-2] => [1,-2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,-1] => [-3,1,2] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,2,-1] => [-3,2,1] => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,3,-2] => [1,-3,2] => 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,-3] => 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,1,-2] => [2,-3,1] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [2,1,-3] => [2,1,-3] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [-4,1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [-4,1,3,2] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [-4,2,1,3] => 1 = 0 + 1
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [-4,2,3,1] => 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [-4,3,1,2] => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [-4,3,2,1] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1,-4,2,3] => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,-4,3,2] => 1 = 0 + 1
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,-4,3] => 1 = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,-4] => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,3,-4,2] => 2 = 1 + 1
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,-4] => 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,-4,1,3] => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [2,-4,3,1] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,-4,3] => 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,-4] => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [2,3,-4,1] => 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [2,3,1,-4] => 3 = 2 + 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [3,-4,1,2] => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [3,-4,2,1] => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [3,1,-4,2] => 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3,1,2,-4] => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [3,2,-4,1] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [3,2,1,-4] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => [-5,1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [-5,1,2,4,3] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [-5,1,3,2,4] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [-5,1,3,4,2] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [-5,1,4,2,3] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [-5,1,4,3,2] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [-5,2,1,3,4] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [-5,2,1,4,3] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [-5,2,3,1,4] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [-5,2,3,4,1] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [-5,2,4,1,3] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [-5,2,4,3,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [-5,3,1,2,4] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [-5,3,1,4,2] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [-5,3,2,1,4] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [-5,3,2,4,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => [-5,3,4,1,2] => 3 = 2 + 1
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [-5,3,4,2,1] => 3 = 2 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,-1] => [-6,1,2,3,4,5] => ? = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,3,4,6,5,-1] => [-6,1,2,3,5,4] => ? = 0 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [2,3,5,4,6,-1] => [-6,1,2,4,3,5] => ? = 0 + 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [2,3,6,4,5,-1] => [-6,1,2,4,5,3] => ? = 0 + 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [2,3,5,6,4,-1] => [-6,1,2,5,3,4] => ? = 1 + 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,3,6,5,4,-1] => [-6,1,2,5,4,3] => ? = 1 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [2,4,3,5,6,-1] => [-6,1,3,2,4,5] => ? = 0 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,4,3,6,5,-1] => [-6,1,3,2,5,4] => ? = 0 + 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [2,5,3,4,6,-1] => [-6,1,3,4,2,5] => ? = 0 + 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [2,6,3,4,5,-1] => [-6,1,3,4,5,2] => ? = 0 + 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [2,5,3,6,4,-1] => [-6,1,3,5,2,4] => ? = 1 + 1
[1,2,4,6,5,3] => [1,2,4,6,5,3] => [2,6,3,5,4,-1] => [-6,1,3,5,4,2] => ? = 1 + 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [2,4,5,3,6,-1] => [-6,1,4,2,3,5] => ? = 1 + 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [2,4,6,3,5,-1] => [-6,1,4,2,5,3] => ? = 1 + 1
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [2,5,4,3,6,-1] => [-6,1,4,3,2,5] => ? = 1 + 1
[1,2,5,4,6,3] => [1,2,5,4,6,3] => [2,6,4,3,5,-1] => [-6,1,4,3,5,2] => ? = 1 + 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [2,5,6,3,4,-1] => [-6,1,4,5,2,3] => ? = 2 + 1
[1,2,5,6,4,3] => [1,2,5,6,4,3] => [2,6,5,3,4,-1] => [-6,1,4,5,3,2] => ? = 2 + 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [2,4,5,6,3,-1] => [-6,1,5,2,3,4] => ? = 1 + 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [2,4,6,5,3,-1] => [-6,1,5,2,4,3] => ? = 1 + 1
[1,2,6,4,3,5] => [1,2,6,4,3,5] => [2,5,4,6,3,-1] => [-6,1,5,3,2,4] => ? = 1 + 1
[1,2,6,4,5,3] => [1,2,6,4,5,3] => [2,6,4,5,3,-1] => [-6,1,5,3,4,2] => ? = 1 + 1
[1,2,6,5,3,4] => [1,2,6,5,3,4] => [2,5,6,4,3,-1] => [-6,1,5,4,2,3] => ? = 1 + 1
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [2,6,5,4,3,-1] => [-6,1,5,4,3,2] => ? = 1 + 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [3,2,4,5,6,-1] => [-6,2,1,3,4,5] => ? = 0 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,2,4,6,5,-1] => [-6,2,1,3,5,4] => ? = 0 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,2,5,4,6,-1] => [-6,2,1,4,3,5] => ? = 0 + 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,2,6,4,5,-1] => [-6,2,1,4,5,3] => ? = 0 + 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,2,5,6,4,-1] => [-6,2,1,5,3,4] => ? = 1 + 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [3,2,6,5,4,-1] => [-6,2,1,5,4,3] => ? = 1 + 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [4,2,3,5,6,-1] => [-6,2,3,1,4,5] => ? = 0 + 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [4,2,3,6,5,-1] => [-6,2,3,1,5,4] => ? = 0 + 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [5,2,3,4,6,-1] => [-6,2,3,4,1,5] => ? = 0 + 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [6,2,3,4,5,-1] => [-6,2,3,4,5,1] => ? = 0 + 1
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [5,2,3,6,4,-1] => [-6,2,3,5,1,4] => ? = 1 + 1
[1,3,4,6,5,2] => [1,3,4,6,5,2] => [6,2,3,5,4,-1] => [-6,2,3,5,4,1] => ? = 1 + 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,2,5,3,6,-1] => [-6,2,4,1,3,5] => ? = 1 + 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [4,2,6,3,5,-1] => [-6,2,4,1,5,3] => ? = 1 + 1
[1,3,5,4,2,6] => [1,3,5,4,2,6] => [5,2,4,3,6,-1] => [-6,2,4,3,1,5] => ? = 1 + 1
[1,3,5,4,6,2] => [1,3,5,4,6,2] => [6,2,4,3,5,-1] => [-6,2,4,3,5,1] => ? = 1 + 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [5,2,6,3,4,-1] => [-6,2,4,5,1,3] => ? = 2 + 1
[1,3,5,6,4,2] => [1,3,5,6,4,2] => [6,2,5,3,4,-1] => [-6,2,4,5,3,1] => ? = 2 + 1
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [4,2,5,6,3,-1] => [-6,2,5,1,3,4] => ? = 1 + 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [4,2,6,5,3,-1] => ? => ? = 1 + 1
[1,3,6,4,2,5] => [1,3,6,4,2,5] => [5,2,4,6,3,-1] => ? => ? = 1 + 1
[1,3,6,4,5,2] => [1,3,6,4,5,2] => [6,2,4,5,3,-1] => [-6,2,5,3,4,1] => ? = 1 + 1
[1,3,6,5,2,4] => [1,3,6,5,2,4] => [5,2,6,4,3,-1] => ? => ? = 1 + 1
[1,3,6,5,4,2] => [1,3,6,5,4,2] => [6,2,5,4,3,-1] => [-6,2,5,4,3,1] => ? = 1 + 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [3,4,2,5,6,-1] => [-6,3,1,2,4,5] => ? = 1 + 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [3,4,2,6,5,-1] => [-6,3,1,2,5,4] => ? = 1 + 1
Description
The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation.
For a signed permutation $\sigma$, this equals
$$ \left\lfloor \dfrac{fexc(\sigma)+1}{2} \right\rfloor = exc(\sigma) + \left\lfloor \dfrac{neg(\sigma)+1}{2} \right\rfloor, $$
where
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|,$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|.$$
This statistic has the same distribution as the descent statistic [[St001427]].
Matching statistic: St000307
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,3,1,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,1,2,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,1,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[3,1,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001632
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,3,1,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,1,2,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,1,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[3,1,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St000632
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 2
[1,4,5,3,2] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 2
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 2
[2,4,5,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 2
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[3,1,2,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 1
[3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 2
[3,1,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
=> ? = 2
[3,2,1,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[3,2,1,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000298
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 2 + 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 2 + 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 2 + 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[3,1,2,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
=> ? = 1 + 1
[3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 2 + 1
[3,1,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
=> ? = 2 + 1
[3,2,1,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[3,2,1,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000640The rank of the largest boolean interval in a poset. St001864The number of excedances of a signed permutation.
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