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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000646
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000646: 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] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 2
[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] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 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: St000710
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 83%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 83%
Values
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,1,2] => [1,3,2] => 0
[1,3,2] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,1,2,3] => [1,4,2,3] => 0
[1,2,4,3] => [4,3,1,2] => [2,4,1,3] => [4,2,1,3] => 1
[1,3,2,4] => [4,2,3,1] => [4,3,1,2] => [3,4,1,2] => 2
[1,3,4,2] => [4,2,1,3] => [3,1,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [4,1,3,2] => [3,4,2,1] => [4,3,2,1] => 1
[1,4,3,2] => [4,1,2,3] => [2,3,4,1] => [3,2,4,1] => 1
[2,1,3,4] => [3,4,2,1] => [4,1,3,2] => [1,4,3,2] => 1
[2,1,4,3] => [3,4,1,2] => [2,4,3,1] => [4,2,3,1] => 1
[2,3,1,4] => [3,2,4,1] => [4,3,2,1] => [3,4,2,1] => 1
[2,3,4,1] => [3,2,1,4] => [3,1,2,4] => [1,3,2,4] => 0
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [4,3,1,2] => 2
[2,4,3,1] => [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[3,1,2,4] => [2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [3,2,4,1] => [2,3,4,1] => 1
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => [2,4,3,1] => 1
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 1
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => [1,5,2,3,4] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,3,4] => [5,2,1,3,4] => 1
[1,2,4,3,5] => [5,4,2,3,1] => [5,3,1,2,4] => [3,5,1,2,4] => 2
[1,2,4,5,3] => [5,4,2,1,3] => [3,1,5,2,4] => [1,3,5,2,4] => 1
[1,2,5,3,4] => [5,4,1,3,2] => [3,5,2,1,4] => [5,3,2,1,4] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [2,3,5,1,4] => [3,2,5,1,4] => 1
[1,3,2,4,5] => [5,3,4,2,1] => [5,1,4,2,3] => [1,5,4,2,3] => 2
[1,3,2,5,4] => [5,3,4,1,2] => [2,5,4,1,3] => [5,2,4,1,3] => 2
[1,3,4,2,5] => [5,3,2,4,1] => [5,4,2,1,3] => [4,5,2,1,3] => 2
[1,3,4,5,2] => [5,3,2,1,4] => [4,1,2,5,3] => [1,4,2,5,3] => 1
[1,3,5,2,4] => [5,3,1,4,2] => [4,5,1,2,3] => [5,4,1,2,3] => 3
[1,3,5,4,2] => [5,3,1,2,4] => [2,4,1,5,3] => [4,2,1,5,3] => 2
[1,4,2,3,5] => [5,2,4,3,1] => [5,4,1,3,2] => [4,5,1,3,2] => 2
[1,4,2,5,3] => [5,2,4,1,3] => [3,4,5,1,2] => [4,3,5,1,2] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,3,4,1,2] => [3,5,4,1,2] => 2
[1,4,3,5,2] => [5,2,3,1,4] => [4,3,1,5,2] => [3,4,1,5,2] => 2
[1,4,5,2,3] => [5,2,1,4,3] => [4,1,5,3,2] => [1,4,5,3,2] => 1
[1,4,5,3,2] => [5,2,1,3,4] => [3,1,4,5,2] => [1,3,4,5,2] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [2,7,1,3,4,5,6] => [7,2,1,3,4,5,6] => ? = 1
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,3,1,2,4,5,6] => [3,7,1,2,4,5,6] => ? = 2
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [3,7,2,1,4,5,6] => [7,3,2,1,4,5,6] => ? = 1
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => [3,2,7,1,4,5,6] => ? = 1
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [7,1,4,2,3,5,6] => [1,7,4,2,3,5,6] => ? = 2
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => [7,2,4,1,3,5,6] => ? = 2
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [7,4,2,1,3,5,6] => [4,7,2,1,3,5,6] => ? = 2
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [4,7,1,2,3,5,6] => [7,4,1,2,3,5,6] => ? = 3
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [2,4,1,7,3,5,6] => [4,2,1,7,3,5,6] => ? = 2
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [7,4,1,3,2,5,6] => [4,7,1,3,2,5,6] => ? = 2
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [3,4,7,1,2,5,6] => [4,3,7,1,2,5,6] => ? = 2
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [7,3,4,1,2,5,6] => [3,7,4,1,2,5,6] => ? = 2
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => [4,3,1,7,2,5,6] => [3,4,1,7,2,5,6] => ? = 2
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [4,1,7,3,2,5,6] => [1,4,7,3,2,5,6] => ? = 1
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [4,7,2,3,1,5,6] => [7,4,2,3,1,5,6] => ? = 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [4,3,7,2,1,5,6] => [3,4,7,2,1,5,6] => ? = 2
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [3,7,4,2,1,5,6] => [7,3,4,2,1,5,6] => ? = 2
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [3,4,2,7,1,5,6] => [4,3,2,7,1,5,6] => ? = 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [2,4,7,3,1,5,6] => [4,2,7,3,1,5,6] => ? = 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [2,3,4,7,1,5,6] => [3,2,4,7,1,5,6] => ? = 1
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [7,1,2,5,3,4,6] => [1,7,2,5,3,4,6] => ? = 2
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [2,7,1,5,3,4,6] => [7,2,1,5,3,4,6] => ? = 3
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [7,3,1,5,2,4,6] => [3,7,1,5,2,4,6] => ? = 3
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [3,7,2,5,1,4,6] => [7,3,2,5,1,4,6] => ? = 2
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => [3,2,7,5,1,4,6] => ? = 2
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,1,5,3,2,4,6] => [1,7,5,3,2,4,6] => ? = 2
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [2,7,5,3,1,4,6] => [7,2,5,3,1,4,6] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [7,5,2,3,1,4,6] => [5,7,2,3,1,4,6] => ? = 2
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [5,1,2,3,7,4,6] => [1,5,2,3,7,4,6] => ? = 1
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [5,7,1,3,2,4,6] => [7,5,1,3,2,4,6] => ? = 3
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [2,5,1,3,7,4,6] => [5,2,1,3,7,4,6] => ? = 2
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [7,5,1,2,3,4,6] => [5,7,1,2,3,4,6] => ? = 4
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [3,5,7,2,1,4,6] => [5,3,7,2,1,4,6] => ? = 3
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => [7,3,5,2,1,4,6] => [3,7,5,2,1,4,6] => ? = 3
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => [5,3,1,2,7,4,6] => [3,5,1,2,7,4,6] => ? = 3
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [5,1,7,2,3,4,6] => [1,5,7,2,3,4,6] => ? = 3
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [5,7,2,1,3,4,6] => [7,5,2,1,3,4,6] => ? = 3
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [5,3,7,1,2,4,6] => [3,5,7,1,2,4,6] => ? = 3
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [3,7,5,1,2,4,6] => [7,3,5,1,2,4,6] => ? = 3
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [3,5,2,1,7,4,6] => [5,3,2,1,7,4,6] => ? = 2
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [2,5,7,1,3,4,6] => [5,2,7,1,3,4,6] => ? = 3
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [2,3,5,1,7,4,6] => [3,2,5,1,7,4,6] => ? = 2
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [7,1,5,2,4,3,6] => [1,7,5,2,4,3,6] => ? = 2
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [2,7,5,1,4,3,6] => [7,2,5,1,4,3,6] => ? = 2
[1,2,5,3,6,4,7] => [7,6,3,5,2,4,1] => [7,4,5,1,2,3,6] => [4,7,5,1,2,3,6] => ? = 3
[1,2,5,3,7,4,6] => [7,6,3,5,1,4,2] => [4,7,5,2,1,3,6] => [7,4,5,2,1,3,6] => ? = 3
[1,2,5,3,7,6,4] => [7,6,3,5,1,2,4] => [2,4,5,7,1,3,6] => [4,2,5,7,1,3,6] => ? = 2
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,1,4,5,2,3,6] => [1,7,4,5,2,3,6] => ? = 2
[1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => [2,7,4,5,1,3,6] => [7,2,4,5,1,3,6] => ? = 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: St000647
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 83%
St000647: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 83%
Values
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => 1
[2,1,3] => [2,3,1] => 1
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => 1
[2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [3,2,4,1] => 1
[2,3,4,1] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,4,1] => 1
[3,2,4,1] => [2,3,1,4] => 1
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => 1
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [5,4,2,3,1] => 2
[1,2,4,5,3] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [5,4,1,3,2] => 1
[1,2,5,4,3] => [5,4,1,2,3] => 1
[1,3,2,4,5] => [5,3,4,2,1] => 2
[1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [5,3,2,4,1] => 2
[1,3,4,5,2] => [5,3,2,1,4] => 1
[1,3,5,2,4] => [5,3,1,4,2] => 3
[1,3,5,4,2] => [5,3,1,2,4] => 2
[1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,2,5,3] => [5,2,4,1,3] => 2
[1,4,3,2,5] => [5,2,3,4,1] => 2
[1,4,3,5,2] => [5,2,3,1,4] => 2
[1,4,5,2,3] => [5,2,1,4,3] => 1
[1,4,5,3,2] => [5,2,1,3,4] => 1
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 2
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 2
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 2
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 1
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 3
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 2
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 2
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 2
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 2
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 2
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 1
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 2
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 2
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 3
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 3
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 2
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 2
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 2
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 2
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 1
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 3
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => ? = 2
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 4
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => ? = 3
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => ? = 3
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => ? = 3
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 3
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 2
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 3
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => ? = 3
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => ? = 3
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => ? = 2
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => ? = 3
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 2
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 2
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 2
[1,2,5,3,6,4,7] => [7,6,3,5,2,4,1] => ? = 3
[1,2,5,3,6,7,4] => [7,6,3,5,2,1,4] => ? = 2
[1,2,5,3,7,4,6] => [7,6,3,5,1,4,2] => ? = 3
[1,2,5,3,7,6,4] => [7,6,3,5,1,2,4] => ? = 2
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 2
[1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 2
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: St000711
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 83%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 83%
Values
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,1,2] => [2,3,1] => 0
[1,3,2] => [3,1,2] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,1,2,3] => [2,3,4,1] => 0
[1,2,4,3] => [4,3,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,3,2,4] => [4,2,3,1] => [4,3,1,2] => [3,4,2,1] => 2
[1,3,4,2] => [4,2,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[1,4,2,3] => [4,1,3,2] => [3,4,2,1] => [4,3,1,2] => 1
[1,4,3,2] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1
[2,1,3,4] => [3,4,2,1] => [4,1,3,2] => [2,4,3,1] => 1
[2,1,4,3] => [3,4,1,2] => [2,4,3,1] => [4,1,3,2] => 1
[2,3,1,4] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 1
[2,3,4,1] => [3,2,1,4] => [3,1,2,4] => [2,3,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 2
[2,4,3,1] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[3,1,2,4] => [2,4,3,1] => [4,2,1,3] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [3,2,4,1] => [4,2,1,3] => 1
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[1,2,4,3,5] => [5,4,2,3,1] => [5,3,1,2,4] => [3,4,2,5,1] => 2
[1,2,4,5,3] => [5,4,2,1,3] => [3,1,5,2,4] => [2,4,1,5,3] => 1
[1,2,5,3,4] => [5,4,1,3,2] => [3,5,2,1,4] => [4,3,1,5,2] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [2,3,5,1,4] => [4,1,2,5,3] => 1
[1,3,2,4,5] => [5,3,4,2,1] => [5,1,4,2,3] => [2,4,5,3,1] => 2
[1,3,2,5,4] => [5,3,4,1,2] => [2,5,4,1,3] => [4,1,5,3,2] => 2
[1,3,4,2,5] => [5,3,2,4,1] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[1,3,4,5,2] => [5,3,2,1,4] => [4,1,2,5,3] => [2,3,5,1,4] => 1
[1,3,5,2,4] => [5,3,1,4,2] => [4,5,1,2,3] => [3,4,5,1,2] => 3
[1,3,5,4,2] => [5,3,1,2,4] => [2,4,1,5,3] => [3,1,5,2,4] => 2
[1,4,2,3,5] => [5,2,4,3,1] => [5,4,1,3,2] => [3,5,4,2,1] => 2
[1,4,2,5,3] => [5,2,4,1,3] => [3,4,5,1,2] => [4,5,1,2,3] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,3,4,1,2] => [4,5,2,3,1] => 2
[1,4,3,5,2] => [5,2,3,1,4] => [4,3,1,5,2] => [3,5,2,1,4] => 2
[1,4,5,2,3] => [5,2,1,4,3] => [4,1,5,3,2] => [2,5,4,1,3] => 1
[1,4,5,3,2] => [5,2,1,3,4] => [3,1,4,5,2] => [2,5,1,3,4] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 1
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [7,3,1,2,4,5,6] => [3,4,2,5,6,7,1] => ? = 2
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [3,1,7,2,4,5,6] => [2,4,1,5,6,7,3] => ? = 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [3,7,2,1,4,5,6] => [4,3,1,5,6,7,2] => ? = 1
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => [4,1,2,5,6,7,3] => ? = 1
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [7,1,4,2,3,5,6] => [2,4,5,3,6,7,1] => ? = 2
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => [4,1,5,3,6,7,2] => ? = 2
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [7,4,2,1,3,5,6] => [4,3,5,2,6,7,1] => ? = 2
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [4,1,2,7,3,5,6] => [2,3,5,1,6,7,4] => ? = 1
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [4,7,1,2,3,5,6] => [3,4,5,1,6,7,2] => ? = 3
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [2,4,1,7,3,5,6] => [3,1,5,2,6,7,4] => ? = 2
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [7,4,1,3,2,5,6] => [3,5,4,2,6,7,1] => ? = 2
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [3,4,7,1,2,5,6] => [4,5,1,2,6,7,3] => ? = 2
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [7,3,4,1,2,5,6] => [4,5,2,3,6,7,1] => ? = 2
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => [4,3,1,7,2,5,6] => [3,5,2,1,6,7,4] => ? = 2
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [4,1,7,3,2,5,6] => [2,5,4,1,6,7,3] => ? = 1
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => [3,1,4,7,2,5,6] => [2,5,1,3,6,7,4] => ? = 1
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [4,7,2,3,1,5,6] => [5,3,4,1,6,7,2] => ? = 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [4,3,7,2,1,5,6] => [5,4,2,1,6,7,3] => ? = 2
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [3,7,4,2,1,5,6] => [5,4,1,3,6,7,2] => ? = 2
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [3,4,2,7,1,5,6] => [5,3,1,2,6,7,4] => ? = 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [2,4,7,3,1,5,6] => [5,1,4,2,6,7,3] => ? = 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [2,3,4,7,1,5,6] => [5,1,2,3,6,7,4] => ? = 1
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [7,1,2,5,3,4,6] => [2,3,5,6,4,7,1] => ? = 2
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [2,7,1,5,3,4,6] => [3,1,5,6,4,7,2] => ? = 3
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [7,3,1,5,2,4,6] => [3,5,2,6,4,7,1] => ? = 3
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => [3,1,7,5,2,4,6] => [2,5,1,6,4,7,3] => ? = 2
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [3,7,2,5,1,4,6] => [5,3,1,6,4,7,2] => ? = 2
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => [5,1,2,6,4,7,3] => ? = 2
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,1,5,3,2,4,6] => [2,5,4,6,3,7,1] => ? = 2
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [2,7,5,3,1,4,6] => [5,1,4,6,3,7,2] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [7,5,2,3,1,4,6] => [5,3,4,6,2,7,1] => ? = 2
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [5,1,2,3,7,4,6] => [2,3,4,6,1,7,5] => ? = 1
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [5,7,1,3,2,4,6] => [3,5,4,6,1,7,2] => ? = 3
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [2,5,1,3,7,4,6] => [3,1,4,6,2,7,5] => ? = 2
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 4
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [3,5,7,2,1,4,6] => [5,4,1,6,2,7,3] => ? = 3
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => [7,3,5,2,1,4,6] => [5,4,2,6,3,7,1] => ? = 3
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => [5,3,1,2,7,4,6] => [3,4,2,6,1,7,5] => ? = 3
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [5,1,7,2,3,4,6] => [2,4,5,6,1,7,3] => ? = 3
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => [3,1,5,2,7,4,6] => [2,4,1,6,3,7,5] => ? = 2
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [5,7,2,1,3,4,6] => [4,3,5,6,1,7,2] => ? = 3
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [5,3,7,1,2,4,6] => [4,5,2,6,1,7,3] => ? = 3
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [3,7,5,1,2,4,6] => [4,5,1,6,3,7,2] => ? = 3
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [3,5,2,1,7,4,6] => [4,3,1,6,2,7,5] => ? = 2
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => ? = 3
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [2,3,5,1,7,4,6] => [4,1,2,6,3,7,5] => ? = 2
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [7,1,5,2,4,3,6] => [2,4,6,5,3,7,1] => ? = 2
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [2,7,5,1,4,3,6] => [4,1,6,5,3,7,2] => ? = 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: St000307
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 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
[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
[1,3,4,2] => [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)
=> ? = 1 + 1
[1,4,2,3] => [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
[1,4,3,2] => [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)
=> ? = 1 + 1
[2,1,3,4] => [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
[2,1,4,3] => [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
[2,3,1,4] => [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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [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
[2,4,3,1] => [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] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,4,2] => [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] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [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)
=> ? = 1 + 1
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[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] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,4,3,5] => [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
[1,2,4,5,3] => [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)
=> ? = 1 + 1
[1,2,5,3,4] => [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
[1,2,5,4,3] => [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)
=> ? = 1 + 1
[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)
=> ? = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 + 1
[1,3,4,2,5] => [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)
=> ? = 2 + 1
[1,3,4,5,2] => [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)
=> ? = 1 + 1
[1,3,5,2,4] => [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)
=> ? = 3 + 1
[1,3,5,4,2] => [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)
=> ? = 2 + 1
[1,4,2,3,5] => [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)
=> ? = 2 + 1
[1,4,2,5,3] => [1,4,2,5,3] => ([(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)
=> ? = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => ([(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)
=> ? = 2 + 1
[1,4,3,5,2] => [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)
=> ? = 2 + 1
[1,4,5,2,3] => [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)
=> ? = 1 + 1
[1,4,5,3,2] => [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)
=> ? = 1 + 1
[1,5,2,3,4] => [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
[1,5,2,4,3] => [1,5,2,4,3] => ([(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)
=> ? = 2 + 1
[1,5,3,2,4] => [1,5,2,4,3] => ([(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)
=> ? = 2 + 1
[1,5,3,4,2] => [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
[1,5,4,2,3] => [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
[1,5,4,3,2] => [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
[2,1,3,4,5] => [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)
=> ? = 1 + 1
[2,1,3,5,4] => [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)
=> ? = 2 + 1
[2,1,4,3,5] => [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)
=> ? = 2 + 1
[2,1,4,5,3] => [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)
=> ? = 1 + 1
[2,1,5,3,4] => [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
[2,1,5,4,3] => [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
[2,3,1,4,5] => [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)
=> ? = 1 + 1
[2,3,1,5,4] => [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
[2,3,4,1,5] => [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
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [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)
=> ? = 2 + 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)
=> ? = 1 + 1
[2,4,1,3,5] => [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)
=> ? = 3 + 1
[2,4,1,5,3] => [1,5,2,4,3] => ([(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)
=> ? = 2 + 1
[2,4,3,1,5] => [1,5,2,4,3] => ([(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)
=> ? = 2 + 1
[2,4,3,5,1] => [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)
=> ? = 2 + 1
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[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
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,2,3,1] => [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
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 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
[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
[1,3,4,2] => [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)
=> ? = 1 + 1
[1,4,2,3] => [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
[1,4,3,2] => [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)
=> ? = 1 + 1
[2,1,3,4] => [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
[2,1,4,3] => [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
[2,3,1,4] => [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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [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
[2,4,3,1] => [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] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,4,2] => [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] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [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)
=> ? = 1 + 1
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[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] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,4,3,5] => [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
[1,2,4,5,3] => [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)
=> ? = 1 + 1
[1,2,5,3,4] => [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
[1,2,5,4,3] => [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)
=> ? = 1 + 1
[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)
=> ? = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 + 1
[1,3,4,2,5] => [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)
=> ? = 2 + 1
[1,3,4,5,2] => [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)
=> ? = 1 + 1
[1,3,5,2,4] => [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)
=> ? = 3 + 1
[1,3,5,4,2] => [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)
=> ? = 2 + 1
[1,4,2,3,5] => [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)
=> ? = 2 + 1
[1,4,2,5,3] => [1,4,2,5,3] => ([(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)
=> ? = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => ([(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)
=> ? = 2 + 1
[1,4,3,5,2] => [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)
=> ? = 2 + 1
[1,4,5,2,3] => [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)
=> ? = 1 + 1
[1,4,5,3,2] => [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)
=> ? = 1 + 1
[1,5,2,3,4] => [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
[1,5,2,4,3] => [1,5,2,4,3] => ([(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)
=> ? = 2 + 1
[1,5,3,2,4] => [1,5,2,4,3] => ([(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)
=> ? = 2 + 1
[1,5,3,4,2] => [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
[1,5,4,2,3] => [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
[1,5,4,3,2] => [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
[2,1,3,4,5] => [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)
=> ? = 1 + 1
[2,1,3,5,4] => [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)
=> ? = 2 + 1
[2,1,4,3,5] => [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)
=> ? = 2 + 1
[2,1,4,5,3] => [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)
=> ? = 1 + 1
[2,1,5,3,4] => [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
[2,1,5,4,3] => [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
[2,3,1,4,5] => [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)
=> ? = 1 + 1
[2,3,1,5,4] => [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
[2,3,4,1,5] => [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
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [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)
=> ? = 2 + 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)
=> ? = 1 + 1
[2,4,1,3,5] => [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)
=> ? = 3 + 1
[2,4,1,5,3] => [1,5,2,4,3] => ([(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)
=> ? = 2 + 1
[2,4,3,1,5] => [1,5,2,4,3] => ([(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)
=> ? = 2 + 1
[2,4,3,5,1] => [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)
=> ? = 2 + 1
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[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
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,2,3,1] => [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: St001330
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,1,3] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,1,2] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,2,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,5,3,2,4] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,5,3,4,2] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,5,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,5,4,3] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,1,4,5] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,1,5,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,4,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,5,1,4] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,4,5,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,4,5,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,5,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,5,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,5,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,5,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,1,2,3,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,2,3,4,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,3,4,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,3,4,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,4,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,4,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,4,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
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