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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000647
(load all 5 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[2,3,1] => [3,1,2] => [3,2,1] => 0 = 1 - 1
[3,1,2] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 0 = 1 - 1
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1 = 2 - 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 1 = 2 - 1
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1 = 2 - 1
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 1 = 2 - 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 2 = 3 - 1
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 2 = 3 - 1
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 1 = 2 - 1
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 0 = 1 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 2 = 3 - 1

Description

The number of big descents of a permutation. For a permutation

$\pi$ , this is the number of indices

$i$ such that

$\pi(i)-\pi(i+1) > 1$ . The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].

Matching statistic: St001907
(load all 2 compositions to match this statistic)

Mapping

Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00162: Signed permutations —inverse⟶ Signed permutations
St001907: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [-1] => [-1] => 1
[1,2] => [1,2] => [2,-1] => [-2,1] => 1
[2,1] => [2,1] => [1,-2] => [1,-2] => 1
[1,2,3] => [1,2,3] => [2,3,-1] => [-3,1,2] => 1
[1,3,2] => [1,3,2] => [3,2,-1] => [-3,2,1] => 1
[2,1,3] => [2,1,3] => [1,3,-2] => [1,-3,2] => 1
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,-3] => 1
[3,1,2] => [3,1,2] => [3,1,-2] => [2,-3,1] => 2
[3,2,1] => [3,2,1] => [2,1,-3] => [2,1,-3] => 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [-4,1,2,3] => 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [-4,1,3,2] => 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [-4,2,1,3] => 1
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [-4,2,3,1] => 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [-4,3,1,2] => 2
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [-4,3,2,1] => 2
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1,-4,2,3] => 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,-4,3,2] => 1
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,-4,3] => 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,-4] => 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,3,-4,2] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,-4] => 2
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,-4,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [2,-4,3,1] => 2
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,-4,3] => 2
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,-4] => 2
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [2,3,-4,1] => 3
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [2,3,1,-4] => 3
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [3,-4,1,2] => 2
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [3,-4,2,1] => 2
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [3,1,-4,2] => 2
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3,1,2,-4] => 2
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [3,2,-4,1] => 2
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [3,2,1,-4] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => [-5,1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [-5,1,2,4,3] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [-5,1,3,2,4] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [-5,1,3,4,2] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [-5,1,4,2,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [-5,1,4,3,2] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [-5,2,1,3,4] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [-5,2,1,4,3] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [-5,2,3,1,4] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [-5,2,3,4,1] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [-5,2,4,1,3] => 2
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [-5,2,4,3,1] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [-5,3,1,2,4] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [-5,3,1,4,2] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [-5,3,2,1,4] => 2
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [-5,3,2,4,1] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => [-5,3,4,1,2] => 3

Description

The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. For a signed permutation

$\sigma$ , this equals

$\left\lfloor \dfrac{fexc(\sigma)+1}{2} \right\rfloor = exc(\sigma) + \left\lfloor \dfrac{neg(\sigma)+1}{2} \right\rfloor,$ where

$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$

$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|,$

$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|.$ This statistic has the same distribution as the descent statistic [[St001427]].

Matching statistic: St000711
(load all 8 compositions to match this statistic)

Mapping

St000711: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1] => ? = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 0 = 1 - 1
[1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 0 = 1 - 1
[2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 1 = 2 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 2 = 3 - 1
[4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 1 = 2 - 1
[4,3,2,1] => 1 = 2 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 0 = 1 - 1
[1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => 1 = 2 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => 0 = 1 - 1
[1,3,4,5,2] => 0 = 1 - 1
[1,3,5,2,4] => 1 = 2 - 1
[1,3,5,4,2] => 1 = 2 - 1
[1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => 1 = 2 - 1
[1,4,3,2,5] => 1 = 2 - 1
[1,4,3,5,2] => 1 = 2 - 1
[1,4,5,2,3] => 2 = 3 - 1
[1,4,5,3,2] => 2 = 3 - 1

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: St000710
(load all 7 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ? = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => 0 = 1 - 1
[2,1,3] => [2,1,3] => 0 = 1 - 1
[2,3,1] => [3,1,2] => 0 = 1 - 1
[3,1,2] => [2,3,1] => 1 = 2 - 1
[3,2,1] => [3,2,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
[1,4,2,3] => [1,3,4,2] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => [3,1,2,4] => 0 = 1 - 1
[2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[2,4,3,1] => [4,1,3,2] => 1 = 2 - 1
[3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => [4,2,1,3] => 1 = 2 - 1
[3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => [4,3,1,2] => 2 = 3 - 1
[4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[4,1,3,2] => [2,4,3,1] => 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => [3,4,2,1] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,4,5,3] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,5,2,3,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,4,2,5,3] => 1 = 2 - 1
[1,3,5,4,2] => [1,5,2,4,3] => 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,3,5,2,4] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,5,3,2,4] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => 2 = 3 - 1
[1,4,5,3,2] => [1,5,4,2,3] => 2 = 3 - 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: St001960
(load all 5 compositions to match this statistic)

Mapping

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => ? = 1 - 1
[1,2] => [2,1] => [2,1] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [2,3,1] => [3,1,2] => 0 = 1 - 1
[1,3,2] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[2,1,3] => [3,2,1] => [2,3,1] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [3,1,2] => [3,2,1] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 0 = 1 - 1
[1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 0 = 1 - 1
[1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[1,4,2,3] => [2,4,1,3] => [4,3,1,2] => 1 = 2 - 1
[1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 0 = 1 - 1
[2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 0 = 1 - 1
[2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[3,1,2,4] => [3,4,2,1] => [4,1,3,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,2,4] => [3,2,1,4] => 1 = 2 - 1
[3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
[3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 2 = 3 - 1
[3,4,2,1] => [1,4,2,3] => [1,4,3,2] => 2 = 3 - 1
[4,1,2,3] => [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[4,1,3,2] => [3,1,4,2] => [4,2,1,3] => 1 = 2 - 1
[4,2,1,3] => [4,3,1,2] => [4,2,3,1] => 1 = 2 - 1
[4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [4,1,3,2] => [3,4,2,1] => 1 = 2 - 1
[4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 1 = 2 - 1
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 0 = 1 - 1
[1,2,4,3,5] => [2,3,5,4,1] => [4,5,1,2,3] => 0 = 1 - 1
[1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [2,3,5,1,4] => [5,4,1,2,3] => 1 = 2 - 1
[1,2,5,4,3] => [2,3,1,5,4] => [3,1,2,5,4] => 1 = 2 - 1
[1,3,2,4,5] => [2,4,3,5,1] => [3,5,1,2,4] => 0 = 1 - 1
[1,3,2,5,4] => [2,4,3,1,5] => [3,4,1,2,5] => 0 = 1 - 1
[1,3,4,2,5] => [2,5,3,4,1] => [3,4,5,1,2] => 0 = 1 - 1
[1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [2,5,3,1,4] => [3,5,4,1,2] => 1 = 2 - 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,1,3,5,4] => 1 = 2 - 1
[1,4,2,3,5] => [2,4,5,3,1] => [5,1,2,4,3] => 1 = 2 - 1
[1,4,2,5,3] => [2,4,1,3,5] => [4,3,1,2,5] => 1 = 2 - 1
[1,4,3,2,5] => [2,5,4,3,1] => [4,3,5,1,2] => 1 = 2 - 1
[1,4,3,5,2] => [2,1,4,3,5] => [2,1,4,3,5] => 1 = 2 - 1
[1,4,5,2,3] => [2,5,1,3,4] => [5,4,3,1,2] => 2 = 3 - 1
[1,4,5,3,2] => [2,1,5,3,4] => [2,1,5,4,3] => 2 = 3 - 1

Description

The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].

Matching statistic: St000646
(load all 3 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[2,3,1] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [2,3,1] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 0 = 1 - 1
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0 = 1 - 1
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 1 = 2 - 1
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => [3,1,2,4] => 1 = 2 - 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 2 = 3 - 1
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0 = 1 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0 = 1 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => [5,1,3,4,2] => 1 = 2 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => [5,2,1,3,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1 = 2 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => [5,1,2,4,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1 = 2 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => [5,3,1,2,4] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => 2 = 3 - 1
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,3,4,2] => [5,1,3,2,4] => 2 = 3 - 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: St001864
(load all 2 compositions to match this statistic)

Mapping

Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00162: Signed permutations —inverse⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 75%

Values

[1] => [1] => [-1] => [-1] => 0 = 1 - 1
[1,2] => [1,2] => [2,-1] => [-2,1] => 0 = 1 - 1
[2,1] => [2,1] => [1,-2] => [1,-2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,-1] => [-3,1,2] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,2,-1] => [-3,2,1] => 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,3,-2] => [1,-3,2] => 0 = 1 - 1
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,-3] => 0 = 1 - 1
[3,1,2] => [3,1,2] => [3,1,-2] => [2,-3,1] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [2,1,-3] => [2,1,-3] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [-4,1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [-4,1,3,2] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [-4,2,1,3] => 0 = 1 - 1
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [-4,2,3,1] => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [-4,3,1,2] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [-4,3,2,1] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1,-4,2,3] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,-4,3,2] => 0 = 1 - 1
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,-4,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,-4] => 0 = 1 - 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,3,-4,2] => 1 = 2 - 1
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,-4] => 1 = 2 - 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,-4,1,3] => 1 = 2 - 1
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [2,-4,3,1] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,-4,3] => 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,-4] => 1 = 2 - 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [2,3,-4,1] => 2 = 3 - 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [2,3,1,-4] => 2 = 3 - 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [3,-4,1,2] => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [3,-4,2,1] => 1 = 2 - 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [3,1,-4,2] => 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3,1,2,-4] => 1 = 2 - 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [3,2,-4,1] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [3,2,1,-4] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => [-5,1,2,3,4] => ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [-5,1,2,4,3] => ? = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [-5,1,3,2,4] => ? = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [-5,1,3,4,2] => ? = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [-5,1,4,2,3] => ? = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [-5,1,4,3,2] => ? = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [-5,2,1,3,4] => ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [-5,2,1,4,3] => ? = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [-5,2,3,1,4] => ? = 1 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [-5,2,3,4,1] => ? = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [-5,2,4,1,3] => ? = 2 - 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [-5,2,4,3,1] => ? = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [-5,3,1,2,4] => ? = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [-5,3,1,4,2] => ? = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [-5,3,2,1,4] => ? = 2 - 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [-5,3,2,4,1] => ? = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => [-5,3,4,1,2] => ? = 3 - 1
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [-5,3,4,2,1] => ? = 3 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [-5,4,1,2,3] => ? = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => [-5,4,1,3,2] => ? = 2 - 1
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => [-5,4,2,1,3] => ? = 2 - 1
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [-5,4,2,3,1] => ? = 2 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [-5,4,3,1,2] => ? = 2 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [-5,4,3,2,1] => ? = 2 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1,-5,2,3,4] => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,-5,2,4,3] => 0 = 1 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,-5,3,2,4] => 0 = 1 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,-5,3,4,2] => 0 = 1 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1,-5,4,2,3] => 1 = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [1,-5,4,3,2] => 1 = 2 - 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,2,-5,3,4] => 0 = 1 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,2,-5,4,3] => 0 = 1 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,2,3,-5,4] => 0 = 1 - 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => [1,2,3,4,-5] => 0 = 1 - 1
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => [1,2,4,-5,3] => 1 = 2 - 1
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [1,2,4,3,-5] => 1 = 2 - 1
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1,3,-5,2,4] => 1 = 2 - 1
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,3,-5,4,2] => 1 = 2 - 1
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [1,3,2,-5,4] => 1 = 2 - 1
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [1,3,2,4,-5] => 1 = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1,3,4,-5,2] => 2 = 3 - 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => [2,-5,1,3,4] => ? = 2 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => [2,-5,1,4,3] => ? = 2 - 1
[3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => [2,-5,3,1,4] => ? = 2 - 1
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [2,-5,3,4,1] => ? = 2 - 1
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => [2,-5,4,1,3] => ? = 3 - 1
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => [2,-5,4,3,1] => ? = 3 - 1
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => [2,1,-5,3,4] => ? = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => [2,1,-5,4,3] => ? = 2 - 1
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => [2,1,3,-5,4] => ? = 2 - 1
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => [2,1,3,4,-5] => ? = 2 - 1
[3,2,5,1,4] => [3,2,5,1,4] => [2,1,5,3,-4] => [2,1,4,-5,3] => ? = 3 - 1
[3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,-5] => [2,1,4,3,-5] => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => [2,3,-5,1,4] => ? = 3 - 1
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [2,3,-5,4,1] => ? = 3 - 1
[3,4,2,1,5] => [3,4,2,1,5] => [3,1,2,5,-4] => [2,3,1,-5,4] => ? = 3 - 1
[3,4,2,5,1] => [3,4,2,5,1] => [3,1,2,4,-5] => [2,3,1,4,-5] => ? = 3 - 1
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => [2,3,4,-5,1] => ? = 4 - 1
[3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,-5] => [2,3,4,1,-5] => ? = 4 - 1
[3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => [2,4,-5,1,3] => ? = 3 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => [2,4,-5,3,1] => ? = 3 - 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => [2,4,1,-5,3] => ? = 3 - 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,4,2,-5] => [2,4,1,3,-5] => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => [2,4,3,-5,1] => ? = 3 - 1
[3,5,4,2,1] => [3,5,4,2,1] => [4,1,3,2,-5] => [2,4,3,1,-5] => ? = 3 - 1
[4,1,2,3,5] => [4,1,2,3,5] => [3,4,1,5,-2] => [3,-5,1,2,4] => ? = 2 - 1
[4,1,2,5,3] => [4,1,2,5,3] => [3,5,1,4,-2] => [3,-5,1,4,2] => ? = 2 - 1

Description

The number of excedances of a signed permutation. For a signed permutation

$\pi\in\mathfrak H_n$ , this is

$\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$ .

Matching statistic: St000307
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => [1] => ([],1)
 => 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
 => 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[3,1,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 3
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 3
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 3
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 3
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,5,3,1,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,1,2,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,1,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 2
[3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,1,5,2,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 3
[3,1,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 3
[3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 2
[3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2

Description

The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset

$P$ . It sends an order ideal

$I$ to the order ideal generated by the minimal antichain of

$P \setminus I$ .

Matching statistic: St001632
(load all 6 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => [1] => ([],1)
 => ? = 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
 => 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[3,1,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 3
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 3
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 3
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 3
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,5,3,1,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,1,2,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,1,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 2
[3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,1,5,2,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 3
[3,1,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 3
[3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 2

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: St000298

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => ([],1)
 => ([],1)
 => 1
[1,2] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,1] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,4,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,3,2,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,5,2,3] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,4,1,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,3,1,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,5,1,3] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,4,5,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,5,4,1,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,1,2,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 3
[3,2,1,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,1,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2

Description

The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.

The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph.