Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000710
(load all 4 compositions to match this statistic)
Mapping
St000710: 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] => 1
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 2
[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] => 1
[1,2,5,3,4] => 0
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[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] => 1
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 5 compositions to match this statistic)
Mapping
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●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] => [2,3,1] => 1
[3,1,2] => [3,2,1] => [3,2,1] => 0
[3,2,1] => [2,3,1] => [3,1,2] => 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] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 0
[1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 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] => [2,3,1,4] => 1
[2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 1
[2,4,1,3] => [4,3,1,2] => [2,4,3,1] => 1
[2,4,3,1] => [3,4,1,2] => [2,4,1,3] => 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,1,4,2] => [4,2,1,3] => [3,4,2,1] => 1
[3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 1
[3,2,4,1] => [2,4,1,3] => [3,4,1,2] => 1
[3,4,1,2] => [3,1,4,2] => [4,2,3,1] => 2
[3,4,2,1] => [4,1,3,2] => [3,2,4,1] => 1
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 0
[4,1,3,2] => [3,4,2,1] => [4,2,1,3] => 1
[4,2,1,3] => [2,4,3,1] => [4,3,1,2] => 1
[4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 1
[4,3,1,2] => [4,2,3,1] => [3,1,4,2] => 2
[4,3,2,1] => [3,2,4,1] => [4,1,3,2] => 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] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 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] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,3,5,4,2] => 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,2,5,3] => [1,5,3,2,4] => [1,4,5,3,2] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,4,5,2,3] => 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,5,3,4,2] => 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,4,3,5,2] => 1
[1,4,2,6,5,7,3] => [1,5,7,3,2,4,6] => [1,4,6,7,3,2,5] => ? = 1
[1,4,2,6,7,3,5] => [1,6,3,2,4,7,5] => [1,4,7,5,6,3,2] => ? = 2
[1,4,2,6,7,5,3] => [1,7,3,2,4,6,5] => [1,4,6,5,7,3,2] => ? = 1
[1,4,2,7,3,6,5] => [1,6,7,5,3,2,4] => [1,4,7,5,3,2,6] => ? = 2
[1,4,2,7,5,3,6] => [1,5,7,6,3,2,4] => [1,4,7,6,3,2,5] => ? = 1
[1,4,2,7,5,6,3] => [1,5,6,7,3,2,4] => [1,4,7,3,2,5,6] => ? = 1
[1,4,2,7,6,5,3] => [1,6,5,7,3,2,4] => [1,4,7,3,2,6,5] => ? = 1
[1,4,3,6,2,7,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[1,4,3,6,7,2,5] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[1,4,3,6,7,5,2] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[1,4,3,7,2,5,6] => [1,3,7,6,5,2,4] => [1,4,7,6,5,2,3] => ? = 1
[1,4,3,7,2,6,5] => [1,3,6,7,5,2,4] => [1,4,7,5,2,3,6] => ? = 2
[1,4,3,7,5,2,6] => [1,3,5,7,6,2,4] => [1,4,7,6,2,3,5] => ? = 1
[1,4,3,7,6,5,2] => [1,3,6,5,7,2,4] => [1,4,7,2,3,6,5] => ? = 1
[1,4,5,2,3,6,7] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 2
[1,4,5,2,3,7,6] => [1,4,2,5,3,7,6] => [1,5,3,4,2,7,6] => ? = 2
[1,4,5,2,6,3,7] => [1,4,2,6,3,5,7] => [1,5,6,3,4,2,7] => ? = 2
[1,4,5,2,6,7,3] => [1,4,2,7,3,5,6] => [1,5,6,7,3,4,2] => ? = 2
[1,4,5,2,7,3,6] => [1,4,2,7,6,3,5] => [1,5,7,6,3,4,2] => ? = 2
[1,4,5,2,7,6,3] => [1,4,2,6,7,3,5] => [1,5,7,3,4,2,6] => ? = 2
[1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => [1,4,6,7,2,5,3] => ? = 2
[1,4,5,6,7,2,3] => [1,6,2,4,7,3,5] => [1,5,7,3,4,6,2] => ? = 2
[1,4,5,6,7,3,2] => [1,7,2,4,6,3,5] => [1,5,6,3,4,7,2] => ? = 2
[1,4,5,7,3,2,6] => [1,5,3,7,6,2,4] => [1,4,7,6,2,5,3] => ? = 2
[1,4,5,7,3,6,2] => [1,5,3,6,7,2,4] => [1,4,7,2,5,3,6] => ? = 2
[1,4,5,7,6,3,2] => [1,6,3,5,7,2,4] => [1,4,7,2,5,6,3] => ? = 2
[1,4,6,2,3,5,7] => [1,4,2,6,5,3,7] => [1,6,5,3,4,2,7] => ? = 2
[1,4,6,2,3,7,5] => [1,4,2,7,5,3,6] => [1,6,7,5,3,4,2] => ? = 3
[1,4,6,2,5,3,7] => [1,4,2,5,6,3,7] => [1,6,3,4,2,5,7] => ? = 2
[1,4,6,2,5,7,3] => [1,4,2,5,7,3,6] => [1,6,7,3,4,2,5] => ? = 2
[1,4,6,2,7,3,5] => [1,4,2,6,3,7,5] => [1,7,5,6,3,4,2] => ? = 3
[1,4,6,2,7,5,3] => [1,4,2,7,3,6,5] => [1,6,5,7,3,4,2] => ? = 2
[1,4,6,5,2,3,7] => [1,5,2,4,6,3,7] => [1,6,3,4,5,2,7] => ? = 2
[1,4,6,5,2,7,3] => [1,5,2,4,7,3,6] => [1,6,7,3,4,5,2] => ? = 2
[1,4,6,5,3,2,7] => [1,6,2,4,5,3,7] => [1,5,3,4,6,2,7] => ? = 2
[1,4,6,5,3,7,2] => [1,7,2,4,5,3,6] => [1,5,3,4,6,7,2] => ? = 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 5 compositions to match this statistic)
Mapping
Mp00066: Permutations —inverse⟶ Permutations
St000711: 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] => 1
[3,1,2] => [2,3,1] => 0
[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] => 1
[1,4,2,3] => [1,3,4,2] => 0
[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] => 1
[2,3,4,1] => [4,1,2,3] => 1
[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] => 0
[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] => 1
[4,1,2,3] => [2,3,4,1] => 0
[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] => 2
[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] => 1
[1,2,5,3,4] => [1,2,4,5,3] => 0
[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] => 1
[1,3,4,5,2] => [1,5,2,3,4] => 1
[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] => 0
[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] => 1
[1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => ? = 1
[1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 1
[1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 1
[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] => ? = 2
[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] => ? = 1
[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] => ? = 2
[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] => ? = 2
[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] => ? = 1
[1,3,5,4,7,6,2] => [1,7,2,4,3,6,5] => ? = 1
[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] => ? = 1
[1,3,5,6,4,7,2] => [1,7,2,5,3,4,6] => ? = 1
[1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 2
[1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => ? = 2
[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] => ? = 1
[1,3,5,7,4,6,2] => [1,7,2,5,3,6,4] => ? = 1
[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] => ? = 2
[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] => ? = 1
[1,3,6,5,2,4,7] => [1,5,2,6,4,3,7] => ? = 2
[1,3,6,5,2,7,4] => [1,5,2,7,4,3,6] => ? = 2
[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] => ? = 3
[1,3,6,7,2,5,4] => [1,5,2,7,6,3,4] => ? = 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: St000646
(load all 6 compositions to match this statistic)
Mapping
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●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] => [3,1,2] => [3,2,1] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[2,3,1] => [3,2,1] => [2,3,1] => [2,1,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => [3,1,2] => [1,3,2] => 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] => [3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 0
[1,3,4,2] => [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 1
[1,4,2,3] => [4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 0
[1,4,3,2] => [4,1,3,2] => [3,4,2,1] => [2,1,3,4] => 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] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 1
[2,3,4,1] => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 1
[2,4,1,3] => [4,2,1,3] => [2,4,3,1] => [3,1,2,4] => 1
[2,4,3,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 1
[3,2,4,1] => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1
[3,4,1,2] => [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 2
[3,4,2,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1
[4,1,2,3] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 0
[4,1,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 1
[4,2,1,3] => [2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 1
[4,2,3,1] => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 1
[4,3,1,2] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2
[4,3,2,1] => [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 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] => [1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 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] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,3,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => [3,4,5,2,1] => 0
[1,3,2,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 0
[1,3,4,2,5] => [3,1,4,2,5] => [4,2,1,3,5] => [2,4,5,3,1] => 1
[1,3,4,5,2] => [3,1,4,5,2] => [5,2,1,3,4] => [1,4,5,3,2] => 1
[1,3,5,2,4] => [3,1,5,2,4] => [5,4,2,1,3] => [1,2,4,5,3] => 1
[1,3,5,4,2] => [3,1,5,4,2] => [4,5,2,1,3] => [2,1,4,5,3] => 1
[1,4,2,3,5] => [4,1,2,3,5] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[1,4,2,5,3] => [4,1,2,5,3] => [5,3,2,1,4] => [1,3,4,5,2] => 1
[1,4,3,2,5] => [4,1,3,2,5] => [3,4,2,1,5] => [3,2,4,5,1] => 1
[1,4,3,5,2] => [4,1,3,5,2] => [3,5,2,1,4] => [3,1,4,5,2] => 1
[1,4,5,2,3] => [4,1,5,2,3] => [4,2,1,5,3] => [2,4,5,1,3] => 2
[1,4,5,3,2] => [4,1,5,3,2] => [5,2,1,4,3] => [1,4,5,2,3] => 1
[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] => [1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 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] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 0
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,5,4,6,7] => [1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 0
[1,2,3,5,4,7,6] => [1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 0
[1,2,3,5,6,4,7] => [1,2,5,3,6,4,7] => [1,2,6,4,3,5,7] => [7,6,2,4,5,3,1] => ? = 1
[1,2,3,5,6,7,4] => [1,2,5,3,6,7,4] => [1,2,7,4,3,5,6] => [7,6,1,4,5,3,2] => ? = 1
[1,2,3,5,7,4,6] => [1,2,5,3,7,4,6] => [1,2,7,6,4,3,5] => [7,6,1,2,4,5,3] => ? = 1
[1,2,3,5,7,6,4] => [1,2,5,3,7,6,4] => [1,2,6,7,4,3,5] => [7,6,2,1,4,5,3] => ? = 1
[1,2,3,6,4,5,7] => [1,2,6,3,4,5,7] => [1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => ? = 0
[1,2,3,6,4,7,5] => [1,2,6,3,4,7,5] => [1,2,7,5,4,3,6] => [7,6,1,3,4,5,2] => ? = 1
[1,2,3,6,5,4,7] => [1,2,6,3,5,4,7] => [1,2,5,6,4,3,7] => [7,6,3,2,4,5,1] => ? = 1
[1,2,3,6,5,7,4] => [1,2,6,3,5,7,4] => [1,2,5,7,4,3,6] => [7,6,3,1,4,5,2] => ? = 1
[1,2,3,6,7,4,5] => [1,2,6,3,7,4,5] => [1,2,6,4,3,7,5] => [7,6,2,4,5,1,3] => ? = 2
[1,2,3,6,7,5,4] => [1,2,6,3,7,5,4] => [1,2,7,4,3,6,5] => [7,6,1,4,5,2,3] => ? = 1
[1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => [1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => ? = 0
[1,2,3,7,4,6,5] => [1,2,7,3,4,6,5] => [1,2,6,7,5,4,3] => [7,6,2,1,3,4,5] => ? = 1
[1,2,3,7,5,4,6] => [1,2,7,3,5,4,6] => [1,2,5,7,6,4,3] => [7,6,3,1,2,4,5] => ? = 1
[1,2,3,7,5,6,4] => [1,2,7,3,5,6,4] => [1,2,5,6,7,4,3] => [7,6,3,2,1,4,5] => ? = 1
[1,2,3,7,6,4,5] => [1,2,7,3,6,4,5] => [1,2,7,5,6,4,3] => [7,6,1,3,2,4,5] => ? = 2
[1,2,3,7,6,5,4] => [1,2,7,3,6,5,4] => [1,2,6,5,7,4,3] => [7,6,2,3,1,4,5] => ? = 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] => [1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 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] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 0
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 1
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 1
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 1
[1,2,4,5,6,3,7] => [1,2,5,4,6,3,7] => [1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 1
[1,2,4,5,6,7,3] => [1,2,5,4,6,7,3] => [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 1
[1,2,4,5,7,3,6] => [1,2,5,4,7,3,6] => [1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => ? = 1
[1,2,4,5,7,6,3] => [1,2,5,4,7,6,3] => [1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 1
[1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => [1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => ? = 1
[1,2,4,6,3,7,5] => [1,2,6,4,3,7,5] => [1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => ? = 2
[1,2,4,6,5,3,7] => [1,2,6,4,5,3,7] => [1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 1
[1,2,4,6,5,7,3] => [1,2,6,4,5,7,3] => [1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 1
[1,2,4,6,7,3,5] => [1,2,6,4,7,3,5] => [1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => ? = 2
[1,2,4,6,7,5,3] => [1,2,6,4,7,5,3] => [1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => ? = 1
[1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => [1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 1
[1,2,4,7,3,6,5] => [1,2,7,4,3,6,5] => [1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 2
[1,2,4,7,5,3,6] => [1,2,7,4,5,3,6] => [1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => ? = 1
[1,2,4,7,5,6,3] => [1,2,7,4,5,6,3] => [1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 1
[1,2,4,7,6,3,5] => [1,2,7,4,6,3,5] => [1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => ? = 2
[1,2,4,7,6,5,3] => [1,2,7,4,6,5,3] => [1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => ? = 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0
[1,2,5,3,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 0
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
Mapping
Mp00066: Permutations —inverse⟶ Permutations
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%
Values
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [3,2,1] => [2,3,1] => 0
[2,3,1] => [3,1,2] => [3,1,2] => [3,2,1] => 1
[3,1,2] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [3,2,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 0
[1,3,4,2] => [1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 1
[1,4,2,3] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[1,4,3,2] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 0
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 0
[2,3,1,4] => [3,1,2,4] => [3,4,2,1] => [4,1,3,2] => 1
[2,3,4,1] => [4,1,2,3] => [3,4,1,2] => [3,1,4,2] => 1
[2,4,1,3] => [3,1,4,2] => [3,1,2,4] => [3,2,1,4] => 1
[2,4,3,1] => [4,1,3,2] => [3,1,4,2] => [4,2,1,3] => 1
[3,1,2,4] => [2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 1
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1
[3,2,4,1] => [4,2,1,3] => [4,3,1,2] => [4,2,3,1] => 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 2
[3,4,2,1] => [4,3,1,2] => [4,1,3,2] => [3,4,2,1] => 1
[4,1,2,3] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,2,1,3] => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,3,1] => [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 1
[4,3,1,2] => [3,4,2,1] => [1,4,2,3] => [1,4,3,2] => 2
[4,3,2,1] => [4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,1,2,3] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [2,3,5,1,4] => [5,4,1,2,3] => 1
[1,2,5,3,4] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => [3,1,2,5,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,1,2,4] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,1,2,5] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [2,4,5,3,1] => [5,1,2,4,3] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,4,5,1,3] => [4,1,2,5,3] => 1
[1,3,5,2,4] => [1,4,2,5,3] => [2,4,1,3,5] => [4,3,1,2,5] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [2,4,1,5,3] => [5,3,1,2,4] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [2,5,3,4,1] => [3,4,5,1,2] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [2,5,3,1,4] => [3,5,4,1,2] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => [4,3,5,1,2] => 1
[1,4,3,5,2] => [1,5,3,2,4] => [2,5,4,1,3] => [5,3,4,1,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,3,4] => [5,4,3,1,2] => 2
[1,4,5,3,2] => [1,5,4,2,3] => [2,5,1,4,3] => [4,5,3,1,2] => 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
[1,2,3,4,6,5] => [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] => [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] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => ? = 1
[1,2,3,6,4,5] => [1,2,3,5,6,4] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => ? = 0
[1,2,3,6,5,4] => [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] => [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] => [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] => [1,2,5,3,4,6] => [2,3,5,6,4,1] => [6,1,2,3,5,4] => ? = 1
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => ? = 1
[1,2,4,6,3,5] => [1,2,5,3,6,4] => [2,3,5,1,4,6] => [5,4,1,2,3,6] => ? = 1
[1,2,4,6,5,3] => [1,2,6,3,5,4] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => ? = 1
[1,2,5,3,4,6] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [4,5,6,1,2,3] => ? = 0
[1,2,5,3,6,4] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [4,6,5,1,2,3] => ? = 1
[1,2,5,4,3,6] => [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] => [1,2,6,4,3,5] => [2,3,6,5,1,4] => [6,4,5,1,2,3] => ? = 1
[1,2,5,6,3,4] => [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] => [1,2,6,5,3,4] => [2,3,6,1,5,4] => [5,6,4,1,2,3] => ? = 1
[1,2,6,3,4,5] => [1,2,4,5,6,3] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => ? = 0
[1,2,6,3,5,4] => [1,2,4,6,5,3] => [2,3,1,4,6,5] => [3,1,2,4,6,5] => ? = 1
[1,2,6,4,3,5] => [1,2,5,4,6,3] => [2,3,1,5,4,6] => [3,1,2,5,4,6] => ? = 1
[1,2,6,4,5,3] => [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] => [1,2,5,6,4,3] => [2,3,1,6,4,5] => [3,1,2,6,5,4] => ? = 2
[1,2,6,5,4,3] => [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] => [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] => [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] => [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] => [1,3,2,6,4,5] => [2,4,3,6,1,5] => [3,6,5,1,2,4] => ? = 1
[1,3,2,6,4,5] => [1,3,2,5,6,4] => [2,4,3,1,5,6] => [3,4,1,2,5,6] => ? = 0
[1,3,2,6,5,4] => [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] => [1,4,2,3,5,6] => [2,4,5,3,6,1] => [6,1,2,4,3,5] => ? = 1
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [2,4,5,3,1,6] => [5,1,2,4,3,6] => ? = 1
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [2,4,5,6,3,1] => [5,3,6,1,2,4] => ? = 1
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => ? = 1
[1,3,4,6,2,5] => [1,5,2,3,6,4] => [2,4,5,1,3,6] => [4,1,2,5,3,6] => ? = 1
[1,3,4,6,5,2] => [1,6,2,3,5,4] => [2,4,5,1,6,3] => [4,1,2,6,3,5] => ? = 1
[1,3,5,2,4,6] => [1,4,2,5,3,6] => [2,4,6,3,5,1] => [5,6,1,2,4,3] => ? = 1
[1,3,5,2,6,4] => [1,4,2,6,3,5] => [2,4,6,3,1,5] => [6,5,1,2,4,3] => ? = 2
[1,3,5,4,2,6] => [1,5,2,4,3,6] => [2,4,6,5,3,1] => [6,1,2,4,5,3] => ? = 1
[1,3,5,4,6,2] => [1,6,2,4,3,5] => [2,4,6,5,1,3] => [5,1,2,4,6,3] => ? = 1
[1,3,5,6,2,4] => [1,5,2,6,3,4] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => ? = 2
[1,3,5,6,4,2] => [1,6,2,5,3,4] => [2,4,6,1,5,3] => [4,1,2,5,6,3] => ? = 1
[1,3,6,2,4,5] => [1,4,2,5,6,3] => [2,4,1,3,5,6] => [4,3,1,2,5,6] => ? = 1
[1,3,6,2,5,4] => [1,4,2,6,5,3] => [2,4,1,3,6,5] => [4,3,1,2,6,5] => ? = 2
[1,3,6,4,2,5] => [1,5,2,4,6,3] => [2,4,1,5,3,6] => [5,3,1,2,4,6] => ? = 1
[1,3,6,4,5,2] => [1,6,2,4,5,3] => [2,4,1,5,6,3] => [6,3,1,2,4,5] => ? = 1
[1,3,6,5,2,4] => [1,5,2,6,4,3] => [2,4,1,6,3,5] => [6,5,3,1,2,4] => ? = 2
[1,3,6,5,4,2] => [1,6,2,5,4,3] => [2,4,1,6,5,3] => [5,6,3,1,2,4] => ? = 1
[1,4,2,3,5,6] => [1,3,4,2,5,6] => [2,5,3,4,6,1] => [3,4,6,1,2,5] => ? = 0
[1,4,2,3,6,5] => [1,3,4,2,6,5] => [2,5,3,4,1,6] => [3,4,5,1,2,6] => ? = 0
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: St000307
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [2,1] => [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] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,3,1] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,2] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [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] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,4,2] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,4,2,3] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [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] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,3,1,4] => [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)
 => ? = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,1,3] => [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)
 => ? = 1 + 1
[2,4,3,1] => [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)
 => ? = 1 + 1
[3,1,2,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,2,1,4] => [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)
 => ? = 1 + 1
[3,2,4,1] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,4,1,2] => [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)
 => ? = 2 + 1
[3,4,2,1] => [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)
 => ? = 1 + 1
[4,1,2,3] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,1,3,2] => [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
[4,2,1,3] => [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)
 => ? = 1 + 1
[4,2,3,1] => [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)
 => ? = 1 + 1
[4,3,1,2] => [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)
 => ? = 2 + 1
[4,3,2,1] => [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)
 => ? = 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] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [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)
 => ? = 1 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,5,4,3] => [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)
 => ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,4,2,5] => [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)
 => ? = 1 + 1
[1,3,4,5,2] => [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)
 => ? = 1 + 1
[1,3,5,2,4] => [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)
 => ? = 1 + 1
[1,3,5,4,2] => [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)
 => ? = 1 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,4,2,5,3] => [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)
 => ? = 1 + 1
[1,4,3,2,5] => [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)
 => ? = 1 + 1
[1,4,3,5,2] => [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)
 => ? = 1 + 1
[1,4,5,2,3] => [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)
 => ? = 2 + 1
[1,4,5,3,2] => [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)
 => ? = 1 + 1
[1,5,2,3,4] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,5,2,4,3] => [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)
 => ? = 1 + 1
[1,5,3,2,4] => [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)
 => ? = 1 + 1
[1,5,3,4,2] => [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)
 => ? = 1 + 1
[1,5,4,2,3] => [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)
 => ? = 2 + 1
[1,5,4,3,2] => [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)
 => ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,4,5,3] => [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)
 => ? = 1 + 1
[2,1,5,3,4] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,5,4,3] => [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)
 => ? = 1 + 1
[2,3,1,4,5] => [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)
 => ? = 1 + 1
[2,3,1,5,4] => [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)
 => ? = 1 + 1
[2,3,4,1,5] => [4,1,2,3,5] => [1,4,3,2,5] => ([(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)
 => ? = 1 + 1
[2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[2,3,5,1,4] => [4,1,2,5,3] => [1,4,5,3,2] => ([(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)
 => ? = 1 + 1
[2,3,5,4,1] => [5,1,2,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1 + 1
[2,4,1,3,5] => [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)
 => ? = 1 + 1
[2,4,1,5,3] => [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)
 => ? = 2 + 1
[2,4,3,1,5] => [4,1,3,2,5] => [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
[2,4,3,5,1] => [5,1,3,2,4] => [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)
 => ? = 1 + 1
[2,4,5,1,3] => [4,1,5,2,3] => [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
[2,4,5,3,1] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1 + 1
[2,5,1,3,4] => [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)
 => ? = 1 + 1
[2,5,1,4,3] => [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)
 => ? = 2 + 1
[2,5,3,1,4] => [4,1,3,5,2] => [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
[2,5,3,4,1] => [5,1,3,4,2] => [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
[2,5,4,1,3] => [4,1,5,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 2 + 1
[2,5,4,3,1] => [5,1,4,3,2] => [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
[3,1,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,2,5,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,4,2,5] => [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)
 => ? = 1 + 1
[3,1,4,5,2] => [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)
 => ? = 1 + 1
[3,1,5,2,4] => [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)
 => ? = 1 + 1
[3,1,5,4,2] => [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)
 => ? = 1 + 1
[4,1,2,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[5,1,2,3,4] => [2,3,4,5,1] => [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] => [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] => [1,2,3,4,6,5] => [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] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,2,3,6,4,5] => [1,2,3,5,6,4] => [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] => [1,2,4,3,5,6] => [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] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,2,5,3,4,6] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,2,6,3,4,5] => [1,2,4,5,6,3] => [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] => [1,3,2,4,5,6] => [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] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,3,2,6,4,5] => [1,3,2,5,6,4] => [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
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [2,1] => [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] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,3,1] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,2] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [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] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,4,2] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,4,2,3] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [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] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,3,1,4] => [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)
 => ? = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[2,4,1,3] => [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)
 => ? = 1 + 1
[2,4,3,1] => [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)
 => ? = 1 + 1
[3,1,2,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,2,1,4] => [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)
 => ? = 1 + 1
[3,2,4,1] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,4,1,2] => [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)
 => ? = 2 + 1
[3,4,2,1] => [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)
 => ? = 1 + 1
[4,1,2,3] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[4,1,3,2] => [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
[4,2,1,3] => [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)
 => ? = 1 + 1
[4,2,3,1] => [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)
 => ? = 1 + 1
[4,3,1,2] => [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)
 => ? = 2 + 1
[4,3,2,1] => [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)
 => ? = 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] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [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)
 => ? = 1 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,5,4,3] => [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)
 => ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,4,2,5] => [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)
 => ? = 1 + 1
[1,3,4,5,2] => [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)
 => ? = 1 + 1
[1,3,5,2,4] => [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)
 => ? = 1 + 1
[1,3,5,4,2] => [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)
 => ? = 1 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,4,2,5,3] => [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)
 => ? = 1 + 1
[1,4,3,2,5] => [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)
 => ? = 1 + 1
[1,4,3,5,2] => [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)
 => ? = 1 + 1
[1,4,5,2,3] => [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)
 => ? = 2 + 1
[1,4,5,3,2] => [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)
 => ? = 1 + 1
[1,5,2,3,4] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,5,2,4,3] => [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)
 => ? = 1 + 1
[1,5,3,2,4] => [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)
 => ? = 1 + 1
[1,5,3,4,2] => [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)
 => ? = 1 + 1
[1,5,4,2,3] => [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)
 => ? = 2 + 1
[1,5,4,3,2] => [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)
 => ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,4,5,3] => [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)
 => ? = 1 + 1
[2,1,5,3,4] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,5,4,3] => [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)
 => ? = 1 + 1
[2,3,1,4,5] => [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)
 => ? = 1 + 1
[2,3,1,5,4] => [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)
 => ? = 1 + 1
[2,3,4,1,5] => [4,1,2,3,5] => [1,4,3,2,5] => ([(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)
 => ? = 1 + 1
[2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[2,3,5,1,4] => [4,1,2,5,3] => [1,4,5,3,2] => ([(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)
 => ? = 1 + 1
[2,3,5,4,1] => [5,1,2,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1 + 1
[2,4,1,3,5] => [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)
 => ? = 1 + 1
[2,4,1,5,3] => [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)
 => ? = 2 + 1
[2,4,3,1,5] => [4,1,3,2,5] => [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
[2,4,3,5,1] => [5,1,3,2,4] => [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)
 => ? = 1 + 1
[2,4,5,1,3] => [4,1,5,2,3] => [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
[2,4,5,3,1] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1 + 1
[2,5,1,3,4] => [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)
 => ? = 1 + 1
[2,5,1,4,3] => [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)
 => ? = 2 + 1
[2,5,3,1,4] => [4,1,3,5,2] => [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
[2,5,3,4,1] => [5,1,3,4,2] => [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
[2,5,4,1,3] => [4,1,5,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 2 + 1
[2,5,4,3,1] => [5,1,4,3,2] => [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
[3,1,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,2,5,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,1,4,2,5] => [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)
 => ? = 1 + 1
[3,1,4,5,2] => [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)
 => ? = 1 + 1
[3,1,5,2,4] => [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)
 => ? = 1 + 1
[3,1,5,4,2] => [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)
 => ? = 1 + 1
[4,1,2,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[5,1,2,3,4] => [2,3,4,5,1] => [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] => [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] => [1,2,3,4,6,5] => [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] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,2,3,6,4,5] => [1,2,3,5,6,4] => [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] => [1,2,4,3,5,6] => [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] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,2,5,3,4,6] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,2,6,3,4,5] => [1,2,4,5,6,3] => [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] => [1,3,2,4,5,6] => [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] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,3,2,6,4,5] => [1,3,2,5,6,4] => [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.