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Your data matches 86 different statistics following compositions of up to 3 maps.
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Matching statistic: St000319
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1,1]
=> [1]
=> 0
[1,3,2] => [1,2] => [1,1]
=> [1]
=> 0
[3,1,2] => [1,2] => [1,1]
=> [1]
=> 0
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4,5] => [2,3,1,4] => [3,1]
=> [1]
=> 0
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1,1]
=> [1]
=> 0
[1,3,2] => [1,2] => [1,1]
=> [1]
=> 0
[3,1,2] => [1,2] => [1,1]
=> [1]
=> 0
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> 0
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4,5] => [2,3,1,4] => [3,1]
=> [1]
=> 0
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001301
(load all 80 compositions to match this statistic)
(load all 80 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,4] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,3,4,2] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,2,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,4,3,2] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,4] => [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
[2,1,4,3] => [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
[2,4,1,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[3,2,1,4] => [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
[3,2,4,1] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,2,1] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[4,1,2,3] => [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
[4,1,3,2] => [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
[4,2,1,3] => [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
[4,3,2,1] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3,5] => [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
[1,2,4,5,3] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,5,3,4] => [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
[1,2,5,4,3] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,2,4,5] => [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
[1,3,2,5,4] => [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
[1,3,4,2,5] => [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
[1,3,4,5,2] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,5,2,4] => [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
[1,3,5,4,2] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,4,2,3,5] => [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
[1,4,2,5,3] => [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
[1,4,3,2,5] => [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
[1,4,3,5,2] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,4,5,2,3] => [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
[1,4,5,3,2] => [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
[1,5,2,3,4] => [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
[1,5,2,4,3] => [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
[1,5,3,2,4] => [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
[1,5,3,4,2] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,5,4,2,3] => [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
[1,5,4,3,2] => [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
[2,1,3,4,5] => [3,2,4,5,1] => [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
[2,1,3,5,4] => [3,2,4,1,5] => [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
[2,1,4,3,5] => [3,2,5,4,1] => [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)
=> ? = 1
[2,1,4,5,3] => [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)
=> ? = 1
[2,1,5,3,4] => [3,2,5,1,4] => [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
[2,1,5,4,3] => [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)
=> ? = 1
[2,3,1,4,5] => [4,2,3,5,1] => [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)
=> ? = 0
[2,3,1,5,4] => [4,2,3,1,5] => [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)
=> ? = 0
[2,3,5,1,4] => [5,2,3,1,4] => [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)
=> ? = 0
[2,4,3,1,5] => [5,2,4,3,1] => [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)
=> ? = 0
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,5,3,1] => [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
[2,5,1,3,4] => [4,2,5,1,3] => [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)
=> ? = 0
[2,5,1,4,3] => [4,2,1,5,3] => [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
[2,5,3,1,4] => [5,2,4,1,3] => [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)
=> ? = 0
[2,5,4,3,1] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[3,1,2,4,5] => [3,4,2,5,1] => [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
[3,1,2,5,4] => [3,4,2,1,5] => [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
[3,1,5,2,4] => [3,5,2,1,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0
[3,2,1,4,5] => [4,3,2,5,1] => [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)
=> ? = 0
[3,2,1,5,4] => [4,3,2,1,5] => [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)
=> ? = 0
[3,2,4,1,5] => [5,3,2,4,1] => [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)
=> ? = 0
[3,2,4,5,1] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,2,5,1,4] => [5,3,2,1,4] => [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)
=> ? = 0
[3,2,5,4,1] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,4,1,2,5] => [4,5,2,3,1] => [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
[3,4,1,5,2] => [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
[3,4,5,1,2] => [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
[3,5,1,2,4] => [4,5,2,1,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0
[3,5,2,1,4] => [5,4,2,1,3] => [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)
=> ? = 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
[3,5,4,1,2] => [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
[4,1,3,2,5] => [3,5,4,2,1] => [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
[4,1,3,5,2] => [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
[4,1,5,3,2] => [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
[4,2,1,3,5] => [4,3,5,2,1] => [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)
=> ? = 0
[4,2,1,5,3] => [4,3,1,2,5] => [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)
=> ? = 0
[4,2,3,5,1] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,4,6,5] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,6,4] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,6,5,4] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,5,6,3] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,4,6,5,3] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,5,4,6,3] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,6,4,5,3] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,5,6,2] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,4,6,5,2] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,5,4,6,2] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,3,6,4,5,2] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,4,3,5,6,2] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,4,3,6,5,2] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,5,3,4,6,2] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,6,3,4,5,2] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,4,3,5,6,1] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[2,4,3,6,5,1] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
Matching statistic: St000908
(load all 80 compositions to match this statistic)
(load all 80 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,3,4,2] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,4,3,2] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [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 = 0 + 1
[2,1,4,3] => [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 = 0 + 1
[2,4,1,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,2,1,4] => [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 = 0 + 1
[3,2,4,1] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[4,1,2,3] => [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)
=> 1 = 0 + 1
[4,1,3,2] => [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)
=> 1 = 0 + 1
[4,2,1,3] => [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 = 0 + 1
[4,3,2,1] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [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 + 1
[1,2,4,5,3] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [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 + 1
[1,2,5,4,3] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [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 + 1
[1,3,2,5,4] => [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 + 1
[1,3,4,2,5] => [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 + 1
[1,3,4,5,2] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [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 + 1
[1,3,5,4,2] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [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 + 1
[1,4,2,5,3] => [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 + 1
[1,4,3,2,5] => [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 + 1
[1,4,3,5,2] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,5,2,3] => [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 + 1
[1,4,5,3,2] => [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 + 1
[1,5,2,3,4] => [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 + 1
[1,5,2,4,3] => [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 + 1
[1,5,3,2,4] => [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 + 1
[1,5,3,4,2] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,5,4,2,3] => [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 + 1
[1,5,4,3,2] => [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 + 1
[2,1,3,4,5] => [3,2,4,5,1] => [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 + 1
[2,1,3,5,4] => [3,2,4,1,5] => [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 + 1
[2,1,4,3,5] => [3,2,5,4,1] => [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)
=> ? = 1 + 1
[2,1,4,5,3] => [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)
=> ? = 1 + 1
[2,1,5,3,4] => [3,2,5,1,4] => [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 + 1
[2,1,5,4,3] => [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)
=> ? = 1 + 1
[2,3,1,4,5] => [4,2,3,5,1] => [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)
=> ? = 0 + 1
[2,3,1,5,4] => [4,2,3,1,5] => [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)
=> ? = 0 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [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)
=> ? = 0 + 1
[2,4,3,1,5] => [5,2,4,3,1] => [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)
=> ? = 0 + 1
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [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 + 1
[2,5,1,3,4] => [4,2,5,1,3] => [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)
=> ? = 0 + 1
[2,5,1,4,3] => [4,2,1,5,3] => [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
[2,5,3,1,4] => [5,2,4,1,3] => [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)
=> ? = 0 + 1
[2,5,4,3,1] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[3,1,2,4,5] => [3,4,2,5,1] => [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 + 1
[3,1,2,5,4] => [3,4,2,1,5] => [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 + 1
[3,1,5,2,4] => [3,5,2,1,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[3,2,1,4,5] => [4,3,2,5,1] => [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)
=> ? = 0 + 1
[3,2,1,5,4] => [4,3,2,1,5] => [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)
=> ? = 0 + 1
[3,2,4,1,5] => [5,3,2,4,1] => [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)
=> ? = 0 + 1
[3,2,4,5,1] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,5,1,4] => [5,3,2,1,4] => [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)
=> ? = 0 + 1
[3,2,5,4,1] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,1,2,5] => [4,5,2,3,1] => [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
[3,4,1,5,2] => [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
[3,4,5,1,2] => [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
[3,5,1,2,4] => [4,5,2,1,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[3,5,2,1,4] => [5,4,2,1,3] => [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)
=> ? = 0 + 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[3,5,4,1,2] => [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
[4,1,3,2,5] => [3,5,4,2,1] => [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 + 1
[4,1,3,5,2] => [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 + 1
[4,1,5,3,2] => [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 + 1
[4,2,1,3,5] => [4,3,5,2,1] => [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)
=> ? = 0 + 1
[4,2,1,5,3] => [4,3,1,2,5] => [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)
=> ? = 0 + 1
[4,2,3,5,1] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,6,5,4] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,6,5,3] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,5,4,6,3] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,6,4,5,3] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,5,6,2] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,6,5,2] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,5,4,6,2] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,6,4,5,2] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,4,3,5,6,2] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,4,3,6,5,2] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,5,3,4,6,2] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,6,3,4,5,2] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,4,6,1] => [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
[2,4,3,5,6,1] => [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
[2,4,3,6,5,1] => [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
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000914
(load all 80 compositions to match this statistic)
(load all 80 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,3,4,2] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,4,3,2] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [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 = 0 + 1
[2,1,4,3] => [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 = 0 + 1
[2,4,1,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,2,1,4] => [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 = 0 + 1
[3,2,4,1] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[4,1,2,3] => [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)
=> 1 = 0 + 1
[4,1,3,2] => [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)
=> 1 = 0 + 1
[4,2,1,3] => [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 = 0 + 1
[4,3,2,1] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [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 + 1
[1,2,4,5,3] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [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 + 1
[1,2,5,4,3] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [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 + 1
[1,3,2,5,4] => [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 + 1
[1,3,4,2,5] => [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 + 1
[1,3,4,5,2] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [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 + 1
[1,3,5,4,2] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [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 + 1
[1,4,2,5,3] => [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 + 1
[1,4,3,2,5] => [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 + 1
[1,4,3,5,2] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,5,2,3] => [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 + 1
[1,4,5,3,2] => [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 + 1
[1,5,2,3,4] => [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 + 1
[1,5,2,4,3] => [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 + 1
[1,5,3,2,4] => [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 + 1
[1,5,3,4,2] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,5,4,2,3] => [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 + 1
[1,5,4,3,2] => [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 + 1
[2,1,3,4,5] => [3,2,4,5,1] => [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 + 1
[2,1,3,5,4] => [3,2,4,1,5] => [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 + 1
[2,1,4,3,5] => [3,2,5,4,1] => [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)
=> ? = 1 + 1
[2,1,4,5,3] => [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)
=> ? = 1 + 1
[2,1,5,3,4] => [3,2,5,1,4] => [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 + 1
[2,1,5,4,3] => [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)
=> ? = 1 + 1
[2,3,1,4,5] => [4,2,3,5,1] => [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)
=> ? = 0 + 1
[2,3,1,5,4] => [4,2,3,1,5] => [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)
=> ? = 0 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [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)
=> ? = 0 + 1
[2,4,3,1,5] => [5,2,4,3,1] => [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)
=> ? = 0 + 1
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [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 + 1
[2,5,1,3,4] => [4,2,5,1,3] => [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)
=> ? = 0 + 1
[2,5,1,4,3] => [4,2,1,5,3] => [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
[2,5,3,1,4] => [5,2,4,1,3] => [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)
=> ? = 0 + 1
[2,5,4,3,1] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[3,1,2,4,5] => [3,4,2,5,1] => [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 + 1
[3,1,2,5,4] => [3,4,2,1,5] => [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 + 1
[3,1,5,2,4] => [3,5,2,1,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
[3,2,1,4,5] => [4,3,2,5,1] => [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)
=> ? = 0 + 1
[3,2,1,5,4] => [4,3,2,1,5] => [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)
=> ? = 0 + 1
[3,2,4,1,5] => [5,3,2,4,1] => [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)
=> ? = 0 + 1
[3,2,4,5,1] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,2,5,1,4] => [5,3,2,1,4] => [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)
=> ? = 0 + 1
[3,2,5,4,1] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,1,2,5] => [4,5,2,3,1] => [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
[3,4,1,5,2] => [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
[3,4,5,1,2] => [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
[3,5,1,2,4] => [4,5,2,1,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
[3,5,2,1,4] => [5,4,2,1,3] => [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)
=> ? = 0 + 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
[3,5,4,1,2] => [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
[4,1,3,2,5] => [3,5,4,2,1] => [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 + 1
[4,1,3,5,2] => [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 + 1
[4,1,5,3,2] => [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 + 1
[4,2,1,3,5] => [4,3,5,2,1] => [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)
=> ? = 0 + 1
[4,2,1,5,3] => [4,3,1,2,5] => [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)
=> ? = 0 + 1
[4,2,3,5,1] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,6,5,4] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,6,5,3] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,5,4,6,3] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,6,4,5,3] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,5,6,2] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,6,5,2] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,5,4,6,2] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,6,4,5,2] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,4,3,5,6,2] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,4,3,6,5,2] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,5,3,4,6,2] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,6,3,4,5,2] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,4,6,1] => [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
[2,4,3,5,6,1] => [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
[2,4,3,6,5,1] => [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
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\
-\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\
0&\text{otherwise}.
\end{cases}
$$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$
\sum_{x\leq y} \mu(x,y).
$$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Matching statistic: St001634
(load all 80 compositions to match this statistic)
(load all 80 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001634: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001634: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> -1 = 0 - 1
[1,3,2] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> -1 = 0 - 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[1,2,3,4] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
[1,2,4,3] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
[1,3,2,4] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
[1,3,4,2] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
[1,4,2,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
[1,4,3,2] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
[2,1,3,4] => [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 = 0 - 1
[2,1,4,3] => [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 = 0 - 1
[2,4,1,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
[3,2,1,4] => [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 = 0 - 1
[3,2,4,1] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
[3,4,2,1] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
[4,1,2,3] => [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)
=> -1 = 0 - 1
[4,1,3,2] => [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)
=> -1 = 0 - 1
[4,2,1,3] => [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 = 0 - 1
[4,3,2,1] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
[1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,2,3,5,4] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,2,4,3,5] => [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 - 1
[1,2,4,5,3] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,2,5,3,4] => [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 - 1
[1,2,5,4,3] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,3,2,4,5] => [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 - 1
[1,3,2,5,4] => [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 - 1
[1,3,4,2,5] => [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 - 1
[1,3,4,5,2] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,3,5,2,4] => [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 - 1
[1,3,5,4,2] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,4,2,3,5] => [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 - 1
[1,4,2,5,3] => [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 - 1
[1,4,3,2,5] => [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 - 1
[1,4,3,5,2] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,4,5,2,3] => [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 - 1
[1,4,5,3,2] => [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 - 1
[1,5,2,3,4] => [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 - 1
[1,5,2,4,3] => [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 - 1
[1,5,3,2,4] => [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 - 1
[1,5,3,4,2] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,5,4,2,3] => [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 - 1
[1,5,4,3,2] => [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 - 1
[2,1,3,4,5] => [3,2,4,5,1] => [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 - 1
[2,1,3,5,4] => [3,2,4,1,5] => [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 - 1
[2,1,4,3,5] => [3,2,5,4,1] => [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)
=> ? = 1 - 1
[2,1,4,5,3] => [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)
=> ? = 1 - 1
[2,1,5,3,4] => [3,2,5,1,4] => [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 - 1
[2,1,5,4,3] => [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)
=> ? = 1 - 1
[2,3,1,4,5] => [4,2,3,5,1] => [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)
=> ? = 0 - 1
[2,3,1,5,4] => [4,2,3,1,5] => [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)
=> ? = 0 - 1
[2,3,5,1,4] => [5,2,3,1,4] => [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)
=> ? = 0 - 1
[2,4,3,1,5] => [5,2,4,3,1] => [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)
=> ? = 0 - 1
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[2,4,5,3,1] => [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 - 1
[2,5,1,3,4] => [4,2,5,1,3] => [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)
=> ? = 0 - 1
[2,5,1,4,3] => [4,2,1,5,3] => [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
[2,5,3,1,4] => [5,2,4,1,3] => [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)
=> ? = 0 - 1
[2,5,4,3,1] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 - 1
[3,1,2,4,5] => [3,4,2,5,1] => [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 - 1
[3,1,2,5,4] => [3,4,2,1,5] => [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 - 1
[3,1,5,2,4] => [3,5,2,1,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 - 1
[3,2,1,4,5] => [4,3,2,5,1] => [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)
=> ? = 0 - 1
[3,2,1,5,4] => [4,3,2,1,5] => [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)
=> ? = 0 - 1
[3,2,4,1,5] => [5,3,2,4,1] => [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)
=> ? = 0 - 1
[3,2,4,5,1] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[3,2,5,1,4] => [5,3,2,1,4] => [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)
=> ? = 0 - 1
[3,2,5,4,1] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[3,4,1,2,5] => [4,5,2,3,1] => [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
[3,4,1,5,2] => [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
[3,4,5,1,2] => [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
[3,5,1,2,4] => [4,5,2,1,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 - 1
[3,5,2,1,4] => [5,4,2,1,3] => [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)
=> ? = 0 - 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
[3,5,4,1,2] => [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
[4,1,3,2,5] => [3,5,4,2,1] => [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 - 1
[4,1,3,5,2] => [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 - 1
[4,1,5,3,2] => [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 - 1
[4,2,1,3,5] => [4,3,5,2,1] => [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)
=> ? = 0 - 1
[4,2,1,5,3] => [4,3,1,2,5] => [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)
=> ? = 0 - 1
[4,2,3,5,1] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,2,3,4,6,5] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,2,3,5,6,4] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,2,3,6,5,4] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,2,4,5,6,3] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,2,4,6,5,3] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,2,5,4,6,3] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,2,6,4,5,3] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,3,4,5,6,2] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,3,4,6,5,2] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,3,5,4,6,2] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,3,6,4,5,2] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,4,3,5,6,2] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,4,3,6,5,2] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,5,3,4,6,2] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[1,6,3,4,5,2] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
[2,3,5,4,6,1] => [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
[2,4,3,5,6,1] => [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
[2,4,3,6,5,1] => [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
Description
The trace of the Coxeter matrix of the incidence algebra of a poset.
Matching statistic: St001771
(load all 237 compositions to match this statistic)
(load all 237 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [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] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,4,2,3] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [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] => [2,1,4,3] => 0
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[3,4,2,1] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,1,2,3] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,3,2] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,2,1,3] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [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] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,2,4,5] => [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] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[1,5,2,3,4] => [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,5,2,4,3] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,5,3,2,4] => [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,5,3,4,2] => [1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[1,5,4,2,3] => [1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,3,4] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,3,5,1,4] => [4,1,2,5,3] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,4,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[2,4,3,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[2,4,5,3,1] => [5,1,4,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 0
[2,5,1,3,4] => [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,5,1,4,3] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,5,3,1,4] => [4,1,3,5,2] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,5,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[3,1,2,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[3,1,2,5,4] => [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,1,5,2,4] => [2,4,1,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 0
[3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
[3,2,5,1,4] => [4,2,1,5,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,2,5,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[3,4,1,2,5] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[3,4,1,5,2] => [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[3,4,5,1,2] => [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[3,5,1,2,4] => [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,5,2,1,4] => [4,3,1,5,2] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,5,2,4,1] => [5,3,1,4,2] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[3,5,4,1,2] => [4,5,1,3,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[4,1,3,2,5] => [2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[4,1,3,5,2] => [2,5,3,1,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[4,1,5,3,2] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
[4,2,1,3,5] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[4,2,1,5,3] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,2,3,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[4,2,3,5,1] => [5,2,3,1,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[4,2,5,1,3] => [4,2,5,1,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,2,5,3,1] => [5,2,4,1,3] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 0
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[4,3,5,2,1] => [5,4,2,1,3] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 1
[4,5,1,3,2] => [3,5,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
[4,5,2,1,3] => [4,3,5,1,2] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,5,2,3,1] => [5,3,4,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 0
[4,5,3,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 1
[5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,2,1,3,4] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[5,2,1,4,3] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[5,2,3,1,4] => [4,2,3,5,1] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[5,2,4,3,1] => [5,2,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[5,3,2,1,4] => [4,3,2,5,1] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[5,3,2,4,1] => [5,3,2,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[5,3,4,1,2] => [4,5,2,3,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[5,4,2,1,3] => [4,3,5,2,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001870
(load all 237 compositions to match this statistic)
(load all 237 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [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] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,4,2,3] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [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] => [2,1,4,3] => 0
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[3,4,2,1] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,1,2,3] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,3,2] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,2,1,3] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [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] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,2,4,5] => [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] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[1,5,2,3,4] => [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,5,2,4,3] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,5,3,2,4] => [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,5,3,4,2] => [1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[1,5,4,2,3] => [1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,3,4] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,3,5,1,4] => [4,1,2,5,3] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,4,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[2,4,3,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[2,4,5,3,1] => [5,1,4,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 0
[2,5,1,3,4] => [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,5,1,4,3] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,5,3,1,4] => [4,1,3,5,2] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,5,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[3,1,2,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[3,1,2,5,4] => [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,1,5,2,4] => [2,4,1,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 0
[3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
[3,2,5,1,4] => [4,2,1,5,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,2,5,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[3,4,1,2,5] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[3,4,1,5,2] => [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[3,4,5,1,2] => [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[3,5,1,2,4] => [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,5,2,1,4] => [4,3,1,5,2] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,5,2,4,1] => [5,3,1,4,2] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[3,5,4,1,2] => [4,5,1,3,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[4,1,3,2,5] => [2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[4,1,3,5,2] => [2,5,3,1,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[4,1,5,3,2] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
[4,2,1,3,5] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[4,2,1,5,3] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,2,3,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[4,2,3,5,1] => [5,2,3,1,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[4,2,5,1,3] => [4,2,5,1,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,2,5,3,1] => [5,2,4,1,3] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 0
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[4,3,5,2,1] => [5,4,2,1,3] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 1
[4,5,1,3,2] => [3,5,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
[4,5,2,1,3] => [4,3,5,1,2] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,5,2,3,1] => [5,3,4,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 0
[4,5,3,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 1
[5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,2,1,3,4] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[5,2,1,4,3] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[5,2,3,1,4] => [4,2,3,5,1] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[5,2,4,3,1] => [5,2,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[5,3,2,1,4] => [4,3,2,5,1] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[5,3,2,4,1] => [5,3,2,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[5,3,4,1,2] => [4,5,2,3,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[5,4,2,1,3] => [4,3,5,2,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 229 compositions to match this statistic)
(load all 229 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [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] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,4,2,3] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [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] => [2,1,4,3] => 0
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[3,4,2,1] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,1,2,3] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,3,2] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,2,1,3] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [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] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,2,4,5] => [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] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[1,5,2,3,4] => [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,5,2,4,3] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,5,3,2,4] => [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,5,3,4,2] => [1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[1,5,4,2,3] => [1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
[2,1,5,3,4] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,3,5,1,4] => [4,1,2,5,3] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,4,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[2,4,3,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[2,4,5,3,1] => [5,1,4,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 0
[2,5,1,3,4] => [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,5,1,4,3] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,5,3,1,4] => [4,1,3,5,2] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[2,5,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[3,1,2,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[3,1,2,5,4] => [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,1,5,2,4] => [2,4,1,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 0
[3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
[3,2,5,1,4] => [4,2,1,5,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,2,5,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[3,4,1,2,5] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[3,4,1,5,2] => [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[3,4,5,1,2] => [4,5,1,2,3] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[3,5,1,2,4] => [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[3,5,2,1,4] => [4,3,1,5,2] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[3,5,2,4,1] => [5,3,1,4,2] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[3,5,4,1,2] => [4,5,1,3,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[4,1,3,2,5] => [2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[4,1,3,5,2] => [2,5,3,1,4] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[4,1,5,3,2] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
[4,2,1,3,5] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[4,2,1,5,3] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,2,3,1,5] => [4,2,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[4,2,3,5,1] => [5,2,3,1,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[4,2,5,1,3] => [4,2,5,1,3] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,2,5,3,1] => [5,2,4,1,3] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 0
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[4,3,5,2,1] => [5,4,2,1,3] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 1
[4,5,1,3,2] => [3,5,4,1,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 0
[4,5,2,1,3] => [4,3,5,1,2] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[4,5,2,3,1] => [5,3,4,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 0
[4,5,3,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 1
[5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,2,1,3,4] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[5,2,1,4,3] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[5,2,3,1,4] => [4,2,3,5,1] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[5,2,4,3,1] => [5,2,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[5,3,2,1,4] => [4,3,2,5,1] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[5,3,2,4,1] => [5,3,2,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[5,3,4,1,2] => [4,5,2,3,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[5,4,2,1,3] => [4,3,5,2,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
Description
The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 20%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
[3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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)
=> ([],1)
=> ? = 0 + 1
[1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([],1)
=> ? = 0 + 1
[1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([],1)
=> ? = 0 + 1
[2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([],1)
=> ? = 0 + 1
[3,2,4,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([],1)
=> ? = 0 + 1
[3,4,2,1] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([],1)
=> ? = 0 + 1
[4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([],1)
=> ? = 0 + 1
[4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([],1)
=> ? = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([],1)
=> ? = 0 + 1
[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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[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)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[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)
=> ([],1)
=> ? = 0 + 1
[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)
=> ([],1)
=> ? = 0 + 1
[1,4,2,3,5] => [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)
=> ? = 0 + 1
[1,4,2,5,3] => [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)
=> ? = 0 + 1
[1,4,3,2,5] => [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),(2,1)],3)
=> 1 = 0 + 1
[1,4,3,5,2] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,4,5,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,4,5,3,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,5,2,3,4] => [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)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[1,5,2,4,3] => [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)
=> ? = 0 + 1
[1,5,3,2,4] => [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)
=> ([(0,1)],2)
=> ? = 0 + 1
[1,5,3,4,2] => [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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,5,4,2,3] => [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)
=> ? = 0 + 1
[1,5,4,3,2] => [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)
=> ([],1)
=> ? = 0 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([],1)
=> ? = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([],1)
=> ? = 0 + 1
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([],1)
=> ? = 1 + 1
[2,3,1,4,5] => [3,1,2,4,5] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([],1)
=> ? = 0 + 1
[2,3,5,1,4] => [5,4,1,2,3] => ([(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)
=> ? = 0 + 1
[2,4,3,1,5] => [3,4,1,2,5] => ([(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)
=> ? = 0 + 1
[2,4,3,5,1] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([],1)
=> ? = 0 + 1
[2,4,5,3,1] => [5,1,2,4,3] => ([(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)
=> ? = 0 + 1
[2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[2,5,1,4,3] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,5,3,1,4] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([],1)
=> ? = 0 + 1
[2,5,4,3,1] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([],1)
=> ? = 0 + 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(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)
=> ? = 0 + 1
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([],1)
=> ? = 0 + 1
[3,1,5,2,4] => [5,4,2,1,3] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([],1)
=> ? = 0 + 1
[3,2,4,1,5] => [2,4,1,3,5] => ([(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)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,2,4,5,1] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([],1)
=> ? = 0 + 1
[3,2,5,1,4] => [2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([],1)
=> ? = 0 + 1
[3,2,5,4,1] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([],1)
=> ? = 0 + 1
[3,4,1,2,5] => [3,1,4,2,5] => ([(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)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,4,1,5,2] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,4,5,1,2] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,5,1,2,4] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([],1)
=> ? = 0 + 1
[3,5,2,1,4] => [5,4,1,3,2] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,5,2,4,1] => [4,5,1,3,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([],1)
=> ? = 0 + 1
[3,5,4,1,2] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([],1)
=> ? = 1 + 1
[4,1,3,2,5] => [3,4,2,1,5] => ([(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)
=> ? = 0 + 1
[4,2,3,1,5] => [2,3,4,1,5] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,5,2,3,1] => [5,1,4,3,2] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,4,3] => [4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[5,1,3,2,4] => [3,5,4,2,1] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[5,1,4,2,3] => [5,3,4,2,1] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[5,1,4,3,2] => [4,3,5,2,1] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[5,2,1,3,4] => [2,5,4,3,1] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[5,3,1,2,4] => [5,4,2,3,1] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[5,4,1,3,2] => [5,2,4,3,1] => ([(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),(2,1)],3)
=> 1 = 0 + 1
[5,4,2,1,3] => [5,3,2,4,1] => ([(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),(2,1)],3)
=> 1 = 0 + 1
[5,4,2,3,1] => [4,3,2,5,1] => ([(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)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,6,3,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,6,3,4,5] => [2,1,6,5,4,3] => ([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,6,1,5] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,6,1,4,5] => [6,5,4,1,2,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3,5,6] => [4,3,2,1,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3,6,5] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[6,1,2,4,5,3] => [4,5,6,3,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[6,1,3,4,5,2] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
The following 76 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000068The number of minimal elements in a poset. St001964The interval resolution global dimension of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001851The number of Hecke atoms of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000098The chromatic number of a graph. St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001654The monophonic hull number of a graph. St001716The 1-improper chromatic number of a graph. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000297The number of leading ones in a binary word. St000627The exponent of a binary word. St001333The cardinality of a minimal edge-isolating set of a graph. St001430The number of positive entries in a signed permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word.
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