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Your data matches 215 different statistics following compositions of up to 3 maps.
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Matching statistic: St000319
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 2
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 3
[3,1]
=> 2
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 4
[4,1]
=> 3
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 5
[5,1]
=> 4
[4,2]
=> 3
[4,1,1]
=> 3
[3,3]
=> 3
[3,2,1]
=> 2
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 6
[6,1]
=> 5
[5,2]
=> 4
[5,1,1]
=> 4
[4,3]
=> 4
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 3
[3,2,2]
=> 2
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 2
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 7
[7,1]
=> 6
[6,2]
=> 5
[6,1,1]
=> 5
[5,3]
=> 5
[5,2,1]
=> 4
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
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 2
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 3
[3,1]
=> 2
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 4
[4,1]
=> 3
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 5
[5,1]
=> 4
[4,2]
=> 3
[4,1,1]
=> 3
[3,3]
=> 3
[3,2,1]
=> 2
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 6
[6,1]
=> 5
[5,2]
=> 4
[5,1,1]
=> 4
[4,3]
=> 4
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 3
[3,2,2]
=> 2
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 2
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 7
[7,1]
=> 6
[6,2]
=> 5
[6,1,1]
=> 5
[5,3]
=> 5
[5,2,1]
=> 4
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: St000394
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,3,3,3,4,4,5,6}
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001726
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 83%●distinct values known / distinct values provided: 70%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001726: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 83%●distinct values known / distinct values provided: 70%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> []
=> []
=> [] => ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4]
=> []
=> []
=> [] => ? = 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[5]
=> []
=> []
=> [] => ? = 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[6]
=> []
=> []
=> [] => ? = 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[7]
=> []
=> []
=> [] => ? ∊ {0,6}
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 4
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,6}
[8]
=> []
=> []
=> [] => ? ∊ {0,6,7}
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 4
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,6,7}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,6,7}
[9]
=> []
=> []
=> [] => ? ∊ {0,6,6,7,8}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,6,6,7,8}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,6,6,7,8}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,6,6,7,8}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,6,6,7,8}
[10]
=> []
=> []
=> [] => ? ∊ {0,6,6,7,7,7,8,9}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,6,6,7,7,7,8,9}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,6,6,7,7,7,8,9}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,6,6,7,7,7,8,9}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,6,6,7,7,7,8,9}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,6,6,7,7,7,8,9}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,6,6,7,7,7,8,9}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,6,6,7,7,7,8,9}
Description
The number of visible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$.
Matching statistic: St001062
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0]
=> [1] => {{1}}
=> ? = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3 = 2 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5 = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}}
=> 1 = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => {{1},{2},{3},{4},{5,6}}
=> 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2 = 1 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => {{1},{2},{3},{4,5,6}}
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => {{1,2,3,4}}
=> 4 = 3 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => {{1},{2},{3,4,5,6}}
=> 4 = 3 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => {{1},{2,3,4,5,6}}
=> 5 = 4 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => {{1,2,3,4,5,6}}
=> 6 = 5 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}}
=> 1 = 0 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => {{1},{2},{3},{4},{5},{6,7}}
=> 2 = 1 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => {{1},{2},{3},{4,6},{5}}
=> 2 = 1 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}}
=> 3 = 2 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => {{1},{2,3,4,5}}
=> 4 = 3 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => {{1},{2},{3,5,6},{4}}
=> 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => {{1},{2},{3},{4,5,6,7}}
=> 4 = 3 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => {{1,2,3,4,5}}
=> 5 = 4 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => {{1},{2,4,5,6},{3}}
=> 4 = 3 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => {{1},{2},{3,4,5,6,7}}
=> 5 = 4 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3 = 2 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => {{1,3,4,5,6},{2}}
=> 5 = 4 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => {{1},{2,3,4,5,6,7}}
=> 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}}
=> 7 = 6 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? ∊ {0,1,1,3,5} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? ∊ {0,1,1,3,5} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => {{1},{2},{3},{4},{5,7},{6}}
=> 2 = 1 + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? ∊ {0,1,1,3,5} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => {{1},{2},{3,4,5,6}}
=> 4 = 3 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => {{1},{2},{3},{4,6,7},{5}}
=> 3 = 2 + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? ∊ {0,1,1,3,5} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 3 = 2 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => {{1},{2,3,4,5,6}}
=> 5 = 4 + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => {{1},{2},{3,6},{4,5}}
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => {{1},{2},{3,5,6,7},{4}}
=> 4 = 3 + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> ? ∊ {0,1,1,3,5} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => {{1},{2},{3},{4},{5},{6},{7,8,9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => {{1},{2},{3},{4},{5,7,8},{6}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,6] => {{1},{2},{3},{4},{5},{6,7,8,9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => {{1},{2},{3},{4,6,7,8},{5}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,5] => {{1},{2},{3},{4},{5,6,7,8,9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => {{1},{2},{3,5,6,7,8},{4}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => {{1},{2},{3},{4,5,6,7,8,9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => {{1},{2},{3,4,5,6,7,8,9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => {{1},{2,3,4,5,6,7,8,9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => {{1,2,3,4,5,6,7,8,9}}
=> ? ∊ {0,1,1,2,2,3,3,4,5,6,6,7,8} + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => {{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => {{1},{2},{3},{4},{5},{6},{7,9},{8}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => {{1},{2},{3},{4},{5},{6},{7},{8,9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,6,5] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => {{1},{2},{3},{4},{5},{6,8,9},{7}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,10,7] => {{1},{2},{3},{4},{5},{6},{7,8,9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,5,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => {{1},{2},{3},{4},{5,8},{6,7}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,4,7,6,8,9,5] => {{1},{2},{3},{4},{5,7,8,9},{6}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,10,6] => {{1},{2},{3},{4},{5},{6,7,8,9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => {{1},{2},{3},{4,7,8},{5,6}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,9,4] => {{1},{2},{3},{4,6,7,8,9},{5}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => {{1},{2},{3},{4},{5,6,7,8,9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => {{1},{2},{3,6,7,8},{4,5}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,9,3] => {{1},{2},{3,5,6,7,8,9},{4}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => {{1},{2},{3},{4,5,6,7,8,9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,4,3,6,7,8,2] => {{1},{2,5,6,7,8},{3,4}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,9,2] => {{1},{2,4,5,6,7,8,9},{3}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,10,3] => {{1},{2},{3,4,5,6,7,8,9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => {{1,3,4,5,6,7,8,9},{2}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,2] => {{1},{2,3,4,5,6,7,8,9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => {{1,2,3,4,5,6,7,8,9,10}}
=> ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,5,5,6,6,7,7,7,7,8,9} + 1
Description
The maximal size of a block of a set partition.
Matching statistic: St001504
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 2 = 0 + 2
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 0 + 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 1 + 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 4 + 2
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 6 = 4 + 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 5 + 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 2 + 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 5 = 3 + 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 6 = 4 + 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6 = 4 + 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 6 = 4 + 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 7 = 5 + 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 8 = 6 + 2
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,2,3,5,5,6,7} + 2
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,2,3,5,5,6,7} + 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,1,2,3,5,5,6,7} + 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,1,2,3,5,5,6,7} + 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 4 = 2 + 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 5 = 3 + 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {0,1,2,3,5,5,6,7} + 2
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,3,5,5,6,7} + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,3,5,5,6,7} + 2
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,2,3,5,5,6,7} + 2
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,3,3,4,5,5,6,6,6,7,8} + 2
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,3,4,4,5,5,5,5,6,6,6,7,7,7,7,7,8,9} + 2
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001727
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 70%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,5}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 4
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,5}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? ∊ {0,1,5,6}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? ∊ {0,1,5,6}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 4
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,1,5,6}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,1,5,6}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => ? ∊ {0,1,1,2,5,5,6,7}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? ∊ {0,1,1,2,5,5,6,7}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? ∊ {0,1,1,2,5,5,6,7}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? ∊ {0,1,1,2,5,5,6,7}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 4
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 5
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 4
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => 3
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,1,1,2,5,5,6,7}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,2,5,5,6,7}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,2,5,5,6,7}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,1,1,2,5,5,6,7}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,1,1,1,2,2,3,5,5,6,6,6,7,8}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [8,4,1,2,3,5,6,7] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,1,5,6,7] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,1,5,6] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,7,1] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => ? ∊ {0,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,6,7,7,7,7,7,8,9}
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St000868
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000868: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 80%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000868: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0]
=> [1] => [1] => 0
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,6,5,4,3] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,5,4,3,1] => 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,6,5,4,3,2] => 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6,5,4,3,2,1] => 5
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,5,6,4] => 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,2,4,6,5,3] => 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,5,4,1] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [1,3,6,5,4,2] => 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,6,5,4,3] => 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,4,1] => 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => [2,6,5,4,3,1] => 4
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [1,7,6,5,4,3,2] => ? ∊ {5,6}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => [7,6,5,4,3,2,1] => ? ∊ {5,6}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {0,1,2,3,4,5,5,6,7}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => ? ∊ {0,1,2,3,4,5,5,6,7}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,7,6] => ? ∊ {0,1,2,3,4,5,5,6,7}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => [1,2,4,5,6,3] => 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,2,3,5,7,6,4] => 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,8,7,6,5] => ? ∊ {0,1,2,3,4,5,5,6,7}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [2,3,4,5,1] => 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => [1,3,4,6,5,2] => 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,2,5,4,6,3] => 3
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,2,4,7,6,5,3] => 3
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [1,2,3,8,7,6,5,4] => ? ∊ {0,1,2,3,4,5,5,6,7}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [2,4,3,5,1] => 3
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [1,2,8,7,6,5,4,3] => ? ∊ {0,1,2,3,4,5,5,6,7}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => [2,7,6,5,4,3,1] => ? ∊ {0,1,2,3,4,5,5,6,7}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [1,8,7,6,5,4,3,2] => ? ∊ {0,1,2,3,4,5,5,6,7}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => ? ∊ {0,1,2,3,4,5,5,6,7}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => [1,2,3,4,5,7,8,6] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,9,8,7] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => [1,2,3,4,6,8,7,5] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,6] => [1,2,3,4,5,9,8,7,6] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => [1,2,3,5,8,7,6,4] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,5] => [1,2,3,4,9,8,7,6,5] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [1,2,4,8,7,6,5,3] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => [1,2,3,9,8,7,6,5,4] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => [2,3,7,6,5,4,1] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => [1,3,8,7,6,5,4,2] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => [1,2,9,8,7,6,5,4,3] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => [3,2,7,6,5,4,1] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => [1,9,8,7,6,5,4,3,2] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [9,8,7,6,5,4,3,2,1] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,7,8}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => [1,2,3,4,5,6,8,9,7] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,10,9,8] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,8,6,7,5] => [1,2,3,4,6,7,8,5] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => [1,2,3,4,5,7,9,8,6] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,10,7] => [1,2,3,4,5,6,10,9,8,7] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,7,5,6,8,4] => [1,2,3,5,6,8,7,4] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => [1,2,3,4,7,6,8,5] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,4,7,6,8,9,5] => [1,2,3,4,6,9,8,7,5] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,10,6] => [1,2,3,4,5,10,9,8,7,6] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,6,4,5,7,8,3] => [1,2,4,5,8,7,6,3] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => [1,2,3,6,5,8,7,4] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,9,4] => [1,2,3,5,9,8,7,6,4] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => [1,2,3,4,10,9,8,7,6,5] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,7,6,5,1] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,5,3,4,6,7,8,2] => [1,3,4,8,7,6,5,2] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => [1,2,5,4,8,7,6,3] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,9,3] => [1,2,4,9,8,7,6,5,3] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => [1,2,3,10,9,8,7,6,5,4] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => [2,4,3,7,6,5,1] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,8,1] => [2,3,8,7,6,5,4,1] => ? ∊ {0,1,1,1,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,9}
Description
The aid statistic in the sense of Shareshian-Wachs.
This is the number of admissible inversions [[St000866]] plus the number of descents [[St000021]]. This statistic was introduced by John Shareshian and Michelle L. Wachs in [1]. Theorem 4.1 states that the aid statistic together with the descent statistic is Euler-Mahonian.
Matching statistic: St000485
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 60%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 60%
Values
[1]
=> [1,0]
=> [1] => ? = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 2 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5 = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3 = 2 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 4 = 3 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 4 = 3 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 5 = 4 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 6 = 5 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 1 = 0 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => 2 = 1 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 2 = 1 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => 3 = 2 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 4 = 3 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => 4 = 3 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 5 = 4 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => 4 = 3 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => 5 = 4 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3 = 2 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 5 = 4 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 6 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => 2 = 1 + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,4,3] => 4 = 3 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => 3 = 2 + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 3 = 2 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => 5 = 4 + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => 4 = 3 + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 5 = 4 + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,1,1,3,5,5,5,6,7} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,6] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,5] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,8,2] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,1,1,2,2,3,3,3,4,4,5,5,6,6,6,6,7,8} + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,10,9] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,8,7] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,6,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,5,8,7,9,6] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,10,7] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,5,8,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,4,7,6,8,9,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,5,7,8,9,10,6] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,4,7,8,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,9,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,3,6,7,8,2] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,9,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,4,2,6,7,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,7,8,9} + 1
Description
The length of the longest cycle of a permutation.
Matching statistic: St001232
(load all 114 compositions to match this statistic)
(load all 114 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 70%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 70%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[5]
=> []
=> []
=> []
=> ? ∊ {0,2}
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,2}
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[6]
=> []
=> []
=> []
=> ? ∊ {0,3,3}
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,3,3}
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {0,3,3}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[7]
=> []
=> []
=> []
=> ? ∊ {0,2,3,4,4}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,2,3,4,4}
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {0,2,3,4,4}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,2,3,4,4}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,2,3,4,4}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[8]
=> []
=> []
=> []
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,3,3,4,4,5,5,5,7}
[9]
=> []
=> []
=> []
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[8,1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,3,3,3,4,4,4,5,5,6,6,6,7,8}
[10]
=> []
=> []
=> []
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[5,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
[3,3,3,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,2,3,3,3,3,4,4,4,5,5,5,5,6,6,6,7,7,7,7,8,9}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
The following 205 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000864The number of circled entries of the shifted recording tableau of a permutation. St000120The number of left tunnels of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000240The number of indices that are not small excedances. St001180Number of indecomposable injective modules with projective dimension at most 1. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000011The number of touch points (or returns) of a Dyck path. St001571The Cartan determinant of the integer partition. St000502The number of successions of a set partitions. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000306The bounce count of a Dyck path. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000006The dinv of a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001717The largest size of an interval in a poset. St000028The number of stack-sorts needed to sort a permutation. St000647The number of big descents of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000031The number of cycles in the cycle decomposition of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000456The monochromatic index of a connected graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001645The pebbling number of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000710The number of big deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000083The number of left oriented leafs of a binary tree except the first one. St000133The "bounce" of a permutation. St000317The cycle descent number of a permutation. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St000719The number of alignments in a perfect matching. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000989The number of final rises of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001684The reduced word complexity of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000015The number of peaks of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000089The absolute variation of a composition. St000155The number of exceedances (also excedences) of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000443The number of long tunnels of a Dyck path. St000536The pathwidth of a graph. St000653The last descent of a permutation. St000702The number of weak deficiencies of a permutation. St000809The reduced reflection length of the permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001589The nesting number of a perfect matching. St000061The number of nodes on the left branch of a binary tree. St000093The cardinality of a maximal independent set of vertices of a graph. St000098The chromatic number of a graph. St000144The pyramid weight of the Dyck path. St000172The Grundy number of a graph. St000676The number of odd rises of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001746The coalition number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001298The number of repeated entries in the Lehmer code of a permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001948The number of augmented double ascents of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001555The order of a signed permutation. St001962The proper pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001330The hat guessing number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001812The biclique partition number of a graph. St001638The book thickness of a graph. St001644The dimension of a graph. St000628The balance of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001488The number of corners of a skew partition. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000736The last entry in the first row of a semistandard tableau. St000181The number of connected components of the Hasse diagram for the poset. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000682The Grundy value of Welter's game on a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001557The number of inversions of the second entry of a permutation. St001712The number of natural descents of a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000766The number of inversions of an integer composition. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001937The size of the center of a parking function. St000822The Hadwiger number of the graph. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000455The second largest eigenvalue of a graph if it is integral. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001875The number of simple modules with projective dimension at most 1. St001960The number of descents of a permutation minus one if its first entry is not one. St000005The bounce statistic of a Dyck path. St000840The number of closers smaller than the largest opener in a perfect matching. St000942The number of critical left to right maxima of the parking functions. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001863The number of weak excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000740The last entry of a permutation. St001424The number of distinct squares in a binary word. St000691The number of changes of a binary word. St000735The last entry on the main diagonal of a standard tableau.
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