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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000319
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => {{1,2}}
=> [2]
=> 1
[1,0,1,0]
=> [3,1,2] => {{1,3},{2}}
=> [2,1]
=> 1
[1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> [3]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> [4]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> [3,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => {{1,5},{2},{3,4}}
=> [2,2,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> [4,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> [3,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> [4,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> [2,2,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> [2,2,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => {{1,4,5},{2,3}}
=> [3,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> [4,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => {{1,2,5},{3,4}}
=> [3,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => {{1,5},{2,3,4}}
=> [3,2]
=> 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> [5]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}}
=> [3,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}}
=> [2,2,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
=> [4,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
=> [4,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}}
=> [2,2,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}}
=> [2,2,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
=> [3,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
=> [4,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
=> [3,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}}
=> [3,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
=> [5,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}}
=> [4,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}}
=> [4,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}}
=> [3,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => {{1,2,4,5,6},{3}}
=> [5,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}}
=> [3,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}}
=> [2,2,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}}
=> [2,2,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}}
=> [3,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}}
=> [3,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}}
=> [2,2,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}}
=> [3,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => {{1,4,5,6},{2,3}}
=> [4,2]
=> 3
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
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => {{1,2}}
=> [2]
=> 1
[1,0,1,0]
=> [3,1,2] => {{1,3},{2}}
=> [2,1]
=> 1
[1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> [3]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> [4]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> [3,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => {{1,5},{2},{3,4}}
=> [2,2,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> [4,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> [3,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> [4,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> [2,2,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> [2,2,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => {{1,4,5},{2,3}}
=> [3,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> [4,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => {{1,2,5},{3,4}}
=> [3,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => {{1,5},{2,3,4}}
=> [3,2]
=> 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> [5]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}}
=> [3,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}}
=> [2,2,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
=> [4,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
=> [4,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}}
=> [2,2,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}}
=> [2,2,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
=> [3,2,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
=> [4,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
=> [3,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}}
=> [3,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
=> [5,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}}
=> [4,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}}
=> [4,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}}
=> [3,2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => {{1,2,4,5,6},{3}}
=> [5,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}}
=> [3,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}}
=> [2,2,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,5},{2,6},{3},{4}}
=> [2,2,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,5,6},{2,4},{3}}
=> [3,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}}
=> [3,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => {{1,6},{2,3},{4,5}}
=> [2,2,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,6},{2,4,5},{3}}
=> [3,2,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => {{1,4,5,6},{2,3}}
=> [4,2]
=> 3
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: St000932
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 96%●distinct values known / distinct values provided: 80%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 96%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8 - 1
[]
=> []
=> []
=> []
=> ? = 0 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9 - 1
[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]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [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]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 5 - 1
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000502
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => {{1,2,3}}
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,2,3,4}}
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,4,5},{2},{3}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,3,4},{2},{5}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,3,5},{2},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,3,4,5},{2}}
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 2 = 3 - 1
[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] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1 - 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] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 2 - 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] => {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 3 - 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] => [1,2,3,4,8,6,7,5] => {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => [1,2,3,4,8,7,6,5] => {{1},{2},{3},{4},{5,8},{6,7}}
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 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] => [8,2,3,4,5,6,7,1] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7,9] => {{1,2,3,4,5,6,7,8},{9}}
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,2,3,4,5,6,7,8] => [1,9,2,3,4,5,6,7,8] => {{1},{2,3,4,5,6,7,8,9}}
=> ? = 8 - 1
[]
=> [] => [] => {}
=> ? = 0 - 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] => [9,2,3,4,5,6,7,8,1] => {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 9 - 1
[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] => {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 1 - 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] => {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,9,7,8] => [1,2,3,4,5,6,9,7,8] => {{1},{2},{3},{4},{5},{6},{7,8,9}}
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [2,1,4,3,6,5,8,7,9] => {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 5 - 1
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: St001189
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St001189: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 94%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
[]
=> ? = 0 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001067
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 94%●distinct values known / distinct values provided: 70%
St001067: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 94%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[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 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[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]
=> ? = 2 - 1
[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]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[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]
=> ? = 7 - 1
[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]
=> ? = 8 - 1
[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]
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8 - 1
[]
=> []
=> ? = 0 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[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]
=> ? = 9 - 1
[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 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 5 - 1
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001640
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0 = 1 - 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]
=> [3,4,1,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3 = 4 - 1
[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]
=> [2,3,4,5,1] => 0 = 1 - 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]
=> [3,4,5,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 2 = 3 - 1
[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]
=> [2,3,4,5,6,7,1] => ? = 1 - 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]
=> [3,4,5,6,7,1,2] => ? = 2 - 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]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 1 - 1
[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]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 3 - 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]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [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]
=> [5,6,7,1,2,3,4] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => ? = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,7,1,2,4,6] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,3] => ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,1,6,7,3,4] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [3,6,1,7,2,4,5] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,7,3,4] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [3,1,4,7,2,5,6] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,1,5,2,6,3,7] => ? = 2 - 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]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,2,3,6,7,4] => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,6,3,7] => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [4,1,2,5,3,7,6] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ? = 4 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [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]
=> [7,1,2,3,4,5,6] => ? = 6 - 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 5 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => ? = 3 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7] => ? = 6 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ? = 1 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,2,3,4] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,4] => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,5,1,6,2,3,7] => ? = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,6,2,3,7,4] => ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,1,7,3,5,6] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,1,7,2,3,5,6] => ? = 4 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,7,3,4] => ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4] => ? = 1 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,1,4,2,7,5,6] => ? = 2 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,2,7,3,4,5] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,5,1,6,2,4,7] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,1,5,6,2,3,7] => ? = 3 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [4,1,5,2,7,3,6] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,4,7] => ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [4,1,5,2,3,7,6] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,7,4,6] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,3,6,7,4] => ? = 2 - 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000441
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 90%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1,0]
=> [.,.]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [[.,.],.]
=> [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [6,5,3,4,1,2,7] => ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> [6,5,2,3,4,1,7] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ? = 3 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [5,3,4,2,1,6,7] => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [7,5,4,3,2,6,1] => ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [7,6,4,2,3,5,1] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [[.,.],[[[.,.],[.,[.,.]]],.]]
=> [6,5,3,4,7,1,2] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[.,.],[[.,[.,[.,.]]],.]],.]
=> [5,4,3,6,1,2,7] => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],[.,[.,.]]]
=> [7,6,3,1,2,4,5] => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [[.,[[[.,.],[.,[.,.]]],.]],.]
=> [5,4,2,3,6,1,7] => ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [[[.,[.,.]],.],[[.,.],[.,.]]]
=> [7,5,6,2,1,3,4] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [[.,[[[.,[.,[.,.]]],.],.]],.]
=> [4,3,2,5,6,1,7] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [6,5,2,1,3,4,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,.]],[[[.,.],.],[.,.]]]
=> [7,4,5,6,2,1,3] => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [7,6,2,3,4,1,5] => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [[[.,[[.,.],.]],[.,[.,.]]],.]
=> [6,5,2,3,1,4,7] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[.,[[[[.,.],.],.],[.,.]]],.]
=> [6,2,3,4,5,1,7] => ? = 4 - 1
[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]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [7,6,4,3,1,2,5] => ? = 2 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [7,5,4,1,2,3,6] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 1 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,[.,.]],[.,.]]],.]
=> [6,4,3,5,1,2,7] => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[[.,.],[.,[.,.]]],.],[.,.]]
=> [7,4,3,1,2,5,6] => ? = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [[.,[[[.,.],[.,.]],[.,.]]],.]
=> [6,4,2,3,5,1,7] => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [5,3,2,4,1,6,7] => ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[[.,.],[[[.,.],[.,.]],.]]]
=> [6,4,5,7,2,3,1] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [[.,[.,[[.,[.,.]],.]]],[.,.]]
=> [7,4,3,5,2,1,6] => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],[[.,[.,.]],.]]
=> [6,5,7,3,1,2,4] => ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [[.,[[.,[[.,[.,.]],.]],.]],.]
=> [4,3,5,2,6,1,7] => ? = 1 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [[[.,[.,.]],.],[[.,[.,.]],.]]
=> [6,5,7,2,1,3,4] => ? = 2 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[[[.,.],.],[[.,[.,.]],.]],.]
=> [5,4,6,1,2,3,7] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,[.,.]],.],[.,.]]]
=> [7,4,3,5,6,1,2] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [[[.,.],[[.,.],[.,.]]],[.,.]]
=> [7,5,3,4,1,2,6] => ? = 3 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [6,4,3,1,2,5,7] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],[.,.]]]
=> [7,4,2,3,5,6,1] => ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [[.,[[[.,[.,.]],.],[.,.]]],.]
=> [6,3,2,4,5,1,7] => ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,[[.,.],[.,.]]],.],[.,.]]
=> [7,4,2,3,1,5,6] => ? = 3 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [[.,[.,.]],[[[.,.],[.,.]],.]]
=> [6,4,5,7,2,1,3] => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [[.,[[.,.],.]],[[.,[.,.]],.]]
=> [6,5,7,2,3,1,4] => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [[[.,[.,.]],[[.,[.,.]],.]],.]
=> [5,4,6,2,1,3,7] => ? = 1 - 1
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000931
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 61%●distinct values known / distinct values provided: 60%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 61%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[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]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2 = 3 - 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]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 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]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 - 1
[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]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 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]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 - 1
[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]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [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]
=> ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 4 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 5 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001223
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 60%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[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 - 1
[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 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,1,1]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6 - 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 5 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 2 - 1
Description
Number 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.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001061The number of indices that are both descents and recoils of a permutation. St000247The number of singleton blocks of a set partition. St000925The number of topologically connected components of a set partition. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000732The number of double deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001948The number of augmented double ascents of a permutation. St000731The number of double exceedences of a permutation. St001330The hat guessing number of a graph. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St000317The cycle descent number of a permutation. 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. St001530The depth of a Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one.
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