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Your data matches 21 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: St001727
(load all 4 compositions to match this statistic)
(load all 4 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] => ? = 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
[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] => ? = 6
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 5
[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] => ? = 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
[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] => ? = 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] => ? = 6
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 5
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? = 5
[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] => ? = 2
[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] => ? = 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] => ? = 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
[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] => ? = 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] => ? = 7
[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] => ? = 6
[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] => ? = 6
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 6
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => ? = 5
[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] => ? = 5
[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] => ? = 3
[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] => ? = 2
[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] => ? = 2
[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] => ? = 1
[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] => ? = 1
[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] => ? = 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
[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] => ? = 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] => ? = 8
[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] => ? = 7
[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] => ? = 7
[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] => ? = 7
[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] => ? = 6
[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] => ? = 6
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => ? = 7
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => ? = 6
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 5
[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] => ? = 5
[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] => ? = 5
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 7
[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] => ? = 4
[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] => ? = 3
[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] => ? = 3
[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] => ? = 3
[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] => ? = 2
[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] => ? = 2
[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] => ? = 2
[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] => ? = 1
[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] => ? = 1
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: St001253
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[4]
=> [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]
=> 3
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[5]
=> [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]
=> 4
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[6]
=> [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]
=> 5
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> 3
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 5
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 4
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> 4
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 3
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 3
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> 3
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 6
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 5
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 4
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 4
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 5
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> 4
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 4
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 7
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 7
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St001278
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0]
=> [.,.]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> [1,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [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]
=> [.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [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]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 5
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 5
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 5
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 4
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 3
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 6
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 5
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 6
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 5
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 4
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 5
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 4
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 4
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 3
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 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]
=> [.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 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]
=> [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 7
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 7
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 7
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 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]
=> [.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6
Description
The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.
Matching statistic: St000502
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 80%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 80%
Values
[1]
=> [[1]]
=> {{1}}
=> {{1}}
=> ? = 0
[2]
=> [[1,2]]
=> {{1,2}}
=> {{1,2}}
=> 1
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> 2
[2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 3
[3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 2
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 1
[2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 4
[4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 3
[3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 5
[5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> 4
[4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> {{1,2,3,4,6},{5}}
=> 3
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> {{1,2,3,4},{5},{6}}
=> 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> {{1,2,3,5,6},{4}}
=> 3
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> {{1,2,3,5},{4},{6}}
=> 2
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> {{1,2,3},{4},{5},{6}}
=> 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> {{1,2,4,6},{3},{5}}
=> 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> {{1,2,4},{3},{5},{6}}
=> 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> {{1,2},{3},{4},{5},{6}}
=> 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> {{1,2,3,4,5,6,7}}
=> 6
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> {{1,2,3,4,5,6},{7}}
=> {{1,2,3,4,5,6},{7}}
=> 5
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> {{1,2,3,4,5},{6,7}}
=> {{1,2,3,4,5,7},{6}}
=> 4
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> {{1,2,3,4,5},{6},{7}}
=> 4
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> {{1,2,3,4,6,7},{5}}
=> 4
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> {{1,2,3,4,6},{5},{7}}
=> 3
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> {{1,2,3,4},{5},{6},{7}}
=> 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> {{1,2,3,5,6},{4},{7}}
=> 3
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> {{1,2,3,5,7},{4},{6}}
=> 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> {{1,2,3,5},{4},{6},{7}}
=> 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> {{1,2,3},{4},{5},{6},{7}}
=> 2
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> {{1,2,4,6},{3},{5},{7}}
=> 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> {{1,2,4},{3},{5},{6},{7}}
=> 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> {{1,2},{3},{4},{5},{6},{7}}
=> {{1,2},{3},{4},{5},{6},{7}}
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> {{1,2,3,4,5,6,7,8}}
=> {{1,2,3,4,5,6,7,8}}
=> 7
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> {{1,2,3,4,5,6,7},{8}}
=> {{1,2,3,4,5,6,7},{8}}
=> 6
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> {{1,2,3,4,5,6},{7,8}}
=> {{1,2,3,4,5,6,8},{7}}
=> 5
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 5
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3,4,5,7,8},{6}}
=> 5
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> {{1,2,3,4,5,7},{6},{8}}
=> ? = 4
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> {{1,2,3,4,5},{6},{7},{8}}
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4,6,7,8},{5}}
=> 5
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> {{1,2,3,4,6,7},{5},{8}}
=> ? = 4
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 3
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> {{1,2,3,4,6},{5},{7},{8}}
=> ? = 3
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 3
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 3
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> {{1,2,3,5,6},{4},{7},{8}}
=> ? = 3
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> {{1,2,3,5,7},{4},{6},{8}}
=> ? = 2
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> {{1,2,3,5},{4},{6},{7},{8}}
=> ? = 2
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> {{1,2,4,6},{3},{5},{7},{8}}
=> ? = 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> {{1,2,3,4,5,6,7,8,9}}
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 8
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> {{1,2,3,4,5,6,7,8},{9}}
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 7
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> {{1,2,3,4,5,6,7},{8,9}}
=> {{1,2,3,4,5,6,7,9},{8}}
=> ? = 6
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> {{1,2,3,4,5,6,7},{8},{9}}
=> {{1,2,3,4,5,6,7},{8},{9}}
=> ? = 6
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> {{1,2,3,4,5,6},{7,8,9}}
=> {{1,2,3,4,5,6,8,9},{7}}
=> ? = 6
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> {{1,2,3,4,5,6},{7,8},{9}}
=> {{1,2,3,4,5,6,8},{7},{9}}
=> ? = 5
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> {{1,2,3,4,5,6},{7},{8},{9}}
=> {{1,2,3,4,5,6},{7},{8},{9}}
=> ? = 5
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> {{1,2,3,4,5},{6,7,8,9}}
=> {{1,2,3,4,5,7,8,9},{6}}
=> ? = 6
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> {{1,2,3,4,5},{6,7,8},{9}}
=> {{1,2,3,4,5,7,8},{6},{9}}
=> ? = 5
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> {{1,2,3,4,5},{6,7},{8,9}}
=> {{1,2,3,4,5,7,9},{6},{8}}
=> ? = 4
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> {{1,2,3,4,5},{6,7},{8},{9}}
=> {{1,2,3,4,5,7},{6},{8},{9}}
=> ? = 4
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 4
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> {{1,2,3,4},{5,6,7,8},{9}}
=> {{1,2,3,4,6,7,8},{5},{9}}
=> ? = 5
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> {{1,2,3,4,6,7,9},{5},{8}}
=> ? = 4
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> {{1,2,3,4,6,7},{5},{8},{9}}
=> ? = 4
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> {{1,2,3,4,6,8},{5},{7},{9}}
=> ? = 3
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> {{1,2,3,4,6},{5},{7},{8},{9}}
=> ? = 3
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> {{1,2,3,4},{5},{6},{7},{8},{9}}
=> {{1,2,3,4},{5},{6},{7},{8},{9}}
=> ? = 3
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> {{1,2,3,5,6,8,9},{4},{7}}
=> ? = 3
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> {{1,2,3,5,6,8},{4},{7},{9}}
=> ? = 3
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> {{1,2,3,5,6},{4},{7},{8},{9}}
=> ? = 3
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> {{1,2,3,5,7,9},{4},{6},{8}}
=> ? = 2
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> {{1,2,3,5,7},{4},{6},{8},{9}}
=> ? = 2
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> {{1,2,3},{4,5},{6},{7},{8},{9}}
=> {{1,2,3,5},{4},{6},{7},{8},{9}}
=> ? = 2
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 2
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> {{1,2,4,6,8},{3},{5},{7},{9}}
=> ? = 1
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> {{1,2,4,6},{3},{5},{7},{8},{9}}
=> ? = 1
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> {{1,2},{3,4},{5},{6},{7},{8},{9}}
=> {{1,2,4},{3},{5},{6},{7},{8},{9}}
=> ? = 1
[2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 9
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> ? = 8
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St000864
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 60%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 60%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [2,1] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [3,2,1] => 2
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,3,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 3
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [3,4,2,1] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 4
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [4,5,3,2,1] => 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,2,5,4,1] => 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,3,4,5,1] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 5
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [5,6,4,3,2,1] => 4
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => 3
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [4,5,6,3,2,1] => 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => 3
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [3,5,2,6,4,1] => 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,4,5,6,2,1] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [2,4,5,1,6,3] => 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [2,3,4,5,6,1] => 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => ? = 5
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [5,4,7,6,3,2,1] => ? = 4
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => ? = 4
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [4,3,2,7,6,5,1] => ? = 4
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [4,6,3,7,5,2,1] => ? = 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => ? = 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [3,6,2,1,7,5,4] => ? = 3
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [3,2,5,4,7,6,1] => ? = 2
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [3,5,6,2,7,4,1] => ? = 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [3,4,5,6,7,2,1] => ? = 2
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,4,1,6,3,7,5] => ? = 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [2,4,5,6,1,7,3] => ? = 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [2,3,4,5,6,7,1] => ? = 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => ? = 7
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [7,8,6,5,4,3,2,1] => ? = 6
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [6,5,8,7,4,3,2,1] => ? = 5
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [6,7,8,5,4,3,2,1] => ? = 5
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [5,4,3,8,7,6,2,1] => ? = 5
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [5,7,4,8,6,3,2,1] => ? = 4
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [5,6,7,8,4,3,2,1] => ? = 4
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => ? = 5
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [4,7,3,2,8,6,5,1] => ? = 4
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [4,3,6,5,8,7,2,1] => ? = 3
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [4,6,7,3,8,5,2,1] => ? = 3
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [4,5,6,7,8,3,2,1] => ? = 3
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [3,2,6,5,1,8,7,4] => ? = 3
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [3,6,7,2,1,8,5,4] => ? = 3
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [3,5,2,7,4,8,6,1] => ? = 2
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [3,5,6,7,2,8,4,1] => ? = 2
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [3,4,5,6,7,8,2,1] => ? = 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => ? = 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [2,4,6,1,7,3,8,5] => ? = 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [2,4,5,6,7,1,8,3] => ? = 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [2,3,4,5,6,7,8,1] => ? = 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 0
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => ? = 8
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [8,9,7,6,5,4,3,2,1] => ? = 7
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [7,6,9,8,5,4,3,2,1] => ? = 6
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [7,8,9,6,5,4,3,2,1] => ? = 6
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [6,5,4,9,8,7,3,2,1] => ? = 6
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [6,8,5,9,7,4,3,2,1] => ? = 5
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [6,7,8,9,5,4,3,2,1] => ? = 5
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [5,4,3,2,9,8,7,6,1] => ? = 6
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [5,8,4,3,9,7,6,2,1] => ? = 5
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [5,4,7,6,9,8,3,2,1] => ? = 4
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [5,7,8,4,9,6,3,2,1] => ? = 4
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [5,6,7,8,9,4,3,2,1] => ? = 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [4,8,3,2,1,9,7,6,5] => ? = 5
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [4,3,7,6,2,9,8,5,1] => ? = 4
Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.
Matching statistic: St001298
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00155: Standard tableaux —promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 70%
Mp00155: Standard tableaux —promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 70%
Values
[1]
=> [[1]]
=> [[1]]
=> [1] => 0
[2]
=> [[1,2]]
=> [[1,2]]
=> [1,2] => 1
[1,1]
=> [[1],[2]]
=> [[1],[2]]
=> [2,1] => 0
[3]
=> [[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => 2
[2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
[3,1]
=> [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3],[2],[4]]
=> [4,2,1,3] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 2
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 5
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 4
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => 3
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => 3
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 3
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => 2
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 6
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 5
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => ? = 4
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,3,4,5,6],[2],[7]]
=> [7,2,1,3,4,5,6] => ? = 4
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => ? = 4
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,3,4,5],[2,7],[6]]
=> [6,2,7,1,3,4,5] => ? = 3
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => ? = 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 3
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => ? = 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,3,4],[2,6],[5],[7]]
=> [7,5,2,6,1,3,4] => ? = 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ? = 2
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? = 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3],[2,5],[4],[6],[7]]
=> [7,6,4,2,5,1,3] => ? = 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => ? = 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 7
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 6
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => ? = 5
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,3,4,5,6,7],[2],[8]]
=> [8,2,1,3,4,5,6,7] => ? = 5
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[1,3,4,5,6],[2,7,8]]
=> [2,7,8,1,3,4,5,6] => ? = 5
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,3,4,5,6],[2,8],[7]]
=> [7,2,8,1,3,4,5,6] => ? = 4
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,3,4,5,6],[2],[7],[8]]
=> [8,7,2,1,3,4,5,6] => ? = 4
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => ? = 5
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,3,4,5],[2,7,8],[6]]
=> [6,2,7,8,1,3,4,5] => ? = 4
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,3,4,5],[2,7],[6,8]]
=> [6,8,2,7,1,3,4,5] => ? = 3
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,3,4,5],[2,7],[6],[8]]
=> [8,6,2,7,1,3,4,5] => ? = 3
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,3,4,5],[2],[6],[7],[8]]
=> [8,7,6,2,1,3,4,5] => ? = 3
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,3,4],[2,6,7],[5,8]]
=> [5,8,2,6,7,1,3,4] => ? = 3
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,3,4],[2,6,7],[5],[8]]
=> [8,5,2,6,7,1,3,4] => ? = 3
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,3,4],[2,6],[5,8],[7]]
=> [7,5,8,2,6,1,3,4] => ? = 2
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[1,3,4],[2,6],[5],[7],[8]]
=> [8,7,5,2,6,1,3,4] => ? = 2
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,3,4],[2],[5],[6],[7],[8]]
=> [8,7,6,5,2,1,3,4] => ? = 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3],[2,5],[4,7],[6,8]]
=> [6,8,4,7,2,5,1,3] => ? = 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[1,3],[2,5],[4,7],[6],[8]]
=> [8,6,4,7,2,5,1,3] => ? = 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[1,3],[2,5],[4],[6],[7],[8]]
=> [8,7,6,4,2,5,1,3] => ? = 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[1,3],[2],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,2,1,3] => ? = 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 0
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => ? = 8
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => ? = 7
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [[1,3,4,5,6,7,8],[2,9]]
=> [2,9,1,3,4,5,6,7,8] => ? = 6
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[1,3,4,5,6,7,8],[2],[9]]
=> [9,2,1,3,4,5,6,7,8] => ? = 6
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [[1,3,4,5,6,7],[2,8,9]]
=> [2,8,9,1,3,4,5,6,7] => ? = 6
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [[1,3,4,5,6,7],[2,9],[8]]
=> [8,2,9,1,3,4,5,6,7] => ? = 5
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,3,4,5,6,7],[2],[8],[9]]
=> [9,8,2,1,3,4,5,6,7] => ? = 5
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [[1,3,4,5,6],[2,7,8,9]]
=> [2,7,8,9,1,3,4,5,6] => ? = 6
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,3,4,5,6],[2,8,9],[7]]
=> [7,2,8,9,1,3,4,5,6] => ? = 5
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,3,4,5,6],[2,8],[7,9]]
=> [7,9,2,8,1,3,4,5,6] => ? = 4
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [[1,3,4,5,6],[2,8],[7],[9]]
=> [9,7,2,8,1,3,4,5,6] => ? = 4
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[1,3,4,5,6],[2],[7],[8],[9]]
=> [9,8,7,2,1,3,4,5,6] => ? = 4
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => ? = 5
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,3,4,5],[2,7,8],[6,9]]
=> [6,9,2,7,8,1,3,4,5] => ? = 4
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Matching statistic: St001232
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 70%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2
[3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[2,1,1,1,1]
=> [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
[1,1,1,1,1,1]
=> [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]
=> 0
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5
[5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 4
[5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 4
[4,2,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 3
[4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3
[3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 2
[3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 2
[3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
[2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[2,1,1,1,1,1]
=> [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
[1,1,1,1,1,1,1]
=> [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]
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[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]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 6
[6,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 5
[5,3]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
[5,2,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 4
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 4
[4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,3,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 4
[4,2,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
[4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 3
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 3
[3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
[3,2,2,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[3,2,1,1,1]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 2
[3,1,1,1,1,1]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
[2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 1
[2,2,1,1,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,1,1,1,1,1,1]
=> [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
[1,1,1,1,1,1,1,1]
=> [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]
=> ? = 0
[9]
=> [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]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 8
[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]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 7
[7,2]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[7,1,1]
=> [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]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? = 6
[6,3]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6
[6,2,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[6,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? = 5
[5,4]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 6
[5,3,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
[5,2,2]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
[5,2,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 4
[5,1,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> ? = 4
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000736
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000736: Semistandard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 60%
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000736: Semistandard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 60%
Values
[1]
=> [[1]]
=> [[1]]
=> [[1]]
=> 1 = 0 + 1
[2]
=> [[1,2]]
=> [[2,0],[1]]
=> [[1,2]]
=> 2 = 1 + 1
[1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> [[1],[2]]
=> 1 = 0 + 1
[3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> 3 = 2 + 1
[2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
[4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5]]
=> 3 = 2 + 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5]]
=> 2 = 1 + 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6]]
=> 6 = 5 + 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6]]
=> ? = 4 + 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6]]
=> ? = 3 + 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5],[6]]
=> ? = 3 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5,6]]
=> ? = 3 + 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5],[6]]
=> ? = 2 + 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4],[5],[6]]
=> ? = 2 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5,6]]
=> ? = 1 + 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5],[6]]
=> ? = 1 + 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4],[5],[6]]
=> ? = 1 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5],[6]]
=> ? = 0 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6,7]]
=> ? = 6 + 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6],[7]]
=> ? = 5 + 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6,7]]
=> ? = 4 + 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6],[7]]
=> ? = 4 + 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6,7]]
=> ? = 4 + 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[4,2,1,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6],[7]]
=> ? = 3 + 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[4,1,1,1,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5],[6],[7]]
=> ? = 3 + 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5,6],[7]]
=> ? = 3 + 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5],[6,7]]
=> ? = 2 + 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[3,2,1,1,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5],[6],[7]]
=> ? = 2 + 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[3,1,1,1,1,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4],[5],[6],[7]]
=> ? = 2 + 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5,6],[7]]
=> ? = 1 + 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[2,2,1,1,1,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5],[6],[7]]
=> ? = 1 + 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[2,1,1,1,1,1,0],[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4],[5],[6],[7]]
=> ? = 1 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> ? = 0 + 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6,7,8]]
=> ? = 7 + 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[7,1,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6,7],[8]]
=> ? = 6 + 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6],[7,8]]
=> ? = 5 + 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[6,1,1,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6],[7],[8]]
=> ? = 5 + 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6,7,8]]
=> ? = 5 + 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[5,2,1,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6,7],[8]]
=> ? = 4 + 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[5,1,1,1,0,0,0,0],[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6],[7],[8]]
=> ? = 4 + 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 5 + 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[4,3,1,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6,7],[8]]
=> ? = 4 + 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[4,2,2,0,0,0,0,0],[4,2,1,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6],[7,8]]
=> ? = 3 + 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[4,2,1,1,0,0,0,0],[4,2,1,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5,6],[7],[8]]
=> ? = 3 + 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[4,1,1,1,1,0,0,0],[4,1,1,1,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> ? = 3 + 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[3,3,2,0,0,0,0,0],[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5,6],[7,8]]
=> ? = 3 + 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[3,3,1,1,0,0,0,0],[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5,6],[7],[8]]
=> ? = 3 + 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[3,2,2,1,0,0,0,0],[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5],[6,7],[8]]
=> ? = 2 + 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[3,2,1,1,1,0,0,0],[3,2,1,1,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> ? = 2 + 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[3,1,1,1,1,1,0,0],[3,1,1,1,1,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> ? = 2 + 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5,6],[7,8]]
=> ? = 1 + 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[2,2,2,1,1,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> ? = 1 + 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[2,2,1,1,1,1,0,0],[2,2,1,1,1,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> ? = 1 + 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[2,1,1,1,1,1,1,0],[2,1,1,1,1,1,0],[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0 + 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6,7,8,9]]
=> ? = 8 + 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [[8,1,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6,7,8],[9]]
=> ? = 7 + 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [[7,2,0,0,0,0,0,0,0],[7,1,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6,7],[8,9]]
=> ? = 6 + 1
Description
The last entry in the first row of a semistandard tableau.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000200The row of the unique '1' in the last column of the alternating sign matrix. 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. St001948The number of augmented double ascents of a permutation. St000005The bounce statistic of a Dyck path. St000840The number of closers smaller than the largest opener in a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001557The number of inversions of the second entry of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000740The last entry of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
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