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Matching statistic: St000319
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> 0
{{1,2}}
=> [2]
=> 1
{{1},{2}}
=> [1,1]
=> 0
{{1,2,3}}
=> [3]
=> 2
{{1,2},{3}}
=> [2,1]
=> 1
{{1,3},{2}}
=> [2,1]
=> 1
{{1},{2,3}}
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> 3
{{1,2,3},{4}}
=> [3,1]
=> 2
{{1,2,4},{3}}
=> [3,1]
=> 2
{{1,2},{3,4}}
=> [2,2]
=> 1
{{1,2},{3},{4}}
=> [2,1,1]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> 2
{{1,3},{2,4}}
=> [2,2]
=> 1
{{1,3},{2},{4}}
=> [2,1,1]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> 1
{{1},{2,3,4}}
=> [3,1]
=> 2
{{1},{2,3},{4}}
=> [2,1,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> 4
{{1,2,3,4},{5}}
=> [4,1]
=> 3
{{1,2,3,5},{4}}
=> [4,1]
=> 3
{{1,2,3},{4,5}}
=> [3,2]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 2
{{1,2,4,5},{3}}
=> [4,1]
=> 3
{{1,2,4},{3,5}}
=> [3,2]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 2
{{1,2,5},{3,4}}
=> [3,2]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> 3
{{1,3,4},{2,5}}
=> [3,2]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 2
{{1,3,5},{2,4}}
=> [3,2]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> 2
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
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> 0
{{1,2}}
=> [2]
=> 1
{{1},{2}}
=> [1,1]
=> 0
{{1,2,3}}
=> [3]
=> 2
{{1,2},{3}}
=> [2,1]
=> 1
{{1,3},{2}}
=> [2,1]
=> 1
{{1},{2,3}}
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> 3
{{1,2,3},{4}}
=> [3,1]
=> 2
{{1,2,4},{3}}
=> [3,1]
=> 2
{{1,2},{3,4}}
=> [2,2]
=> 1
{{1,2},{3},{4}}
=> [2,1,1]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> 2
{{1,3},{2,4}}
=> [2,2]
=> 1
{{1,3},{2},{4}}
=> [2,1,1]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> 1
{{1},{2,3,4}}
=> [3,1]
=> 2
{{1},{2,3},{4}}
=> [2,1,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> 4
{{1,2,3,4},{5}}
=> [4,1]
=> 3
{{1,2,3,5},{4}}
=> [4,1]
=> 3
{{1,2,3},{4,5}}
=> [3,2]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 2
{{1,2,4,5},{3}}
=> [4,1]
=> 3
{{1,2,4},{3,5}}
=> [3,2]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 2
{{1,2,5},{3,4}}
=> [3,2]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> 3
{{1,3,4},{2,5}}
=> [3,2]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 2
{{1,3,5},{2,4}}
=> [3,2]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> 2
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: St000877
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [[1]]
=> => ? = 0
{{1,2}}
=> [2]
=> [[1,2]]
=> 0 => 1
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> 1 => 0
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> 00 => 2
{{1,2},{3}}
=> [2,1]
=> [[1,2],[3]]
=> 01 => 1
{{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> 01 => 1
{{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> 01 => 1
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 0
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> 000 => 3
{{1,2,3},{4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 001 => 2
{{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> 001 => 2
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 1
{{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> 001 => 2
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 1
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
{{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 001 => 2
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 0
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 4
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 3
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 3
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 2
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 3
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 2
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 1
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 3
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 2
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 1
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0010 => 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St001176
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 100%●distinct values known / distinct values provided: 90%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 100%●distinct values known / distinct values provided: 90%
Values
{{1}}
=> [1]
=> [1]
=> []
=> ? = 0 - 1
{{1,2}}
=> [2]
=> [1,1]
=> [1]
=> 0 = 1 - 1
{{1},{2}}
=> [1,1]
=> [2]
=> []
=> ? = 0 - 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> [1]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> []
=> ? = 0 - 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> [2]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> []
=> ? = 0 - 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3 = 4 - 1
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 0 = 1 - 1
{{1,5},{2,3,4}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
{{1},{2,3,4,5}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 0 - 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [6]
=> []
=> ? = 0 - 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1]
=> [7]
=> []
=> ? = 0 - 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1]
=> [8]
=> []
=> ? = 0 - 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> ? = 0 - 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> ? = 0 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001727
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 3
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 4
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 5
{{1,2,3,4,5,6,7}}
=> [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
{{1,2,3,4,5,6},{7}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 5
{{1,2,3,4,5,7},{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
{{1,2,3,4,6,7},{5}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 5
{{1,2,3,5,6,7},{4}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 5
{{1,2,4,5,6,7},{3}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 5
{{1,3,4,5,6,7},{2}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 5
{{1},{2,3,4,5,6,7}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 5
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 0
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [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},{2},{3},{4},{5},{6,8},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [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},{2},{3},{4},{5,7},{6},{8}}
=> [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},{2},{3},{4},{5,8},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
{{1},{2},{3,4,5,6,7,8}}
=> [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
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
{{1},{2,4,5,6,7,8},{3}}
=> [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
{{1},{2,3,4,6,7,8},{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
{{1},{2,3,4,5,7,8},{6}}
=> [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
{{1},{2,3,4,5,6,8},{7}}
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? = 5
{{1},{2,3,4,5,6,7,8}}
=> [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
{{1,2},{3},{4},{5},{6},{7},{8}}
=> [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
{{1,2},{3,4,5,6,7,8}}
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 5
{{1,3},{2},{4},{5},{6},{7},{8}}
=> [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
{{1,4},{2},{3},{5},{6},{7},{8}}
=> [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
{{1,5},{2},{3},{4},{6},{7},{8}}
=> [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
{{1,6},{2},{3},{4},{5},{7},{8}}
=> [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
{{1,7},{2},{3},{4},{5},{6},{8}}
=> [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
{{1,8},{2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
{{1,3,4,5,6,7},{2},{8}}
=> [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
{{1,3,4,5,6,8},{2},{7}}
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? = 5
{{1,3,4,5,6,7,8},{2}}
=> [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
{{1,4,5,6,7,8},{2,3}}
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 5
{{1,2,4,6,7,8},{3},{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
{{1,2,4,5,7,8},{3},{6}}
=> [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
{{1,2,4,5,6,8},{3},{7}}
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? = 5
{{1,2,4,5,6,7,8},{3}}
=> [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
{{1,2,3,6,7,8},{4},{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
{{1,2,3,5,7,8},{4},{6}}
=> [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
{{1,2,3,5,6,7},{4},{8}}
=> [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
{{1,2,3,5,6,7,8},{4}}
=> [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
{{1,2,3,6,7,8},{4,5}}
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 5
{{1,2,3,4,7,8},{5},{6}}
=> [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
{{1,2,3,4,6,7},{5},{8}}
=> [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
{{1,2,3,4,6,8},{5},{7}}
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? = 5
{{1,2,3,4,6,7,8},{5}}
=> [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
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: St001205
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001205: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001205: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2,3,4,5,6,7}}
=> [7] => [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]
=> ? = 6
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
Description
The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Nakayama algebra and the relation to Dyck paths.
Matching statistic: St001233
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
{{1,2,3,4,5,6,7}}
=> [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 5
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 5
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 4
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 5
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 4
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 4
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 5
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 4
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 4
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 4
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 4
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 3
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 3
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 3
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
Description
The number of indecomposable 2-dimensional modules with projective dimension one.
Matching statistic: St001024
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> 1 = 0 + 1
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2,3,4,5,6,7}}
=> [7] => [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]
=> ? = 6 + 1
{{1,2,3,4,5,6},{7}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 + 1
{{1,2,3,4,5,7},{6}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 + 1
{{1,2,3,4,5},{6,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 + 1
{{1,2,3,4,6,7},{5}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 + 1
{{1,2,3,4,6},{5,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 + 1
{{1,2,3,4,7},{5,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 + 1
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
{{1,2,3,5,6,7},{4}}
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 + 1
{{1,2,3,5,6},{4,7}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 + 1
{{1,2,3,5,7},{4,6}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 + 1
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
{{1,2,3,6,7},{4,5}}
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 + 1
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 + 1
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000028
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [2,1] => 1 = 0 + 1
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4 = 3 + 1
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3 = 2 + 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3 = 2 + 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3 = 2 + 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 1 + 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5 = 4 + 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4 = 3 + 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4 = 3 + 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 3 = 2 + 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4 = 3 + 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 3 = 2 + 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 3 = 2 + 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 3 = 2 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4 = 3 + 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 3 = 2 + 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 3 = 2 + 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 3 = 2 + 1
{{1,2,3},{4,5,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 3 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 2 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 2 + 1
{{1,2,4},{3,5,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 3 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,2,4},{3},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 2 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 2 + 1
{{1,2,5},{3,4,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 3 + 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,2,6},{3,4,5}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 3 + 1
{{1,2},{3,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 3 + 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 1
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 1
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 + 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,2,5},{3},{4,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 2 + 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 2 + 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 + 1
{{1,2,6},{3},{4,5}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 2 + 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 1
{{1,2},{3},{4,5,6}}
=> [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 2 + 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 1 + 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 2 + 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 + 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 1 + 1
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 1 + 1
{{1,3,4},{2,5,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 3 + 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,3,4},{2},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 2 + 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 2 + 1
{{1,3,5},{2,4,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 3 + 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,3,6},{2,4,5}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 3 + 1
{{1,3},{2,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 3 + 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 1
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2 + 1
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 + 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 2 + 1
{{1,3,5},{2},{4,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 2 + 1
Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 80%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
{{1,2}}
=> [2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
{{1},{2}}
=> [1,1] => [2] => ([],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
{{1,2},{3}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
{{1},{2,3}}
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1] => [3] => ([],3)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,2,4},{3}}
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => ([],4)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,3},{2,4,5}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,4},{2,3,5}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,5},{2,3,4}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1},{2,3,4,5}}
=> [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
{{1},{2,3,4},{5}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2,3,5},{4}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,4,5},{2},{3}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2,4,5},{3}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2},{3,4}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
{{1},{2},{3,4,5}}
=> [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [5] => ([],5)
=> 1 = 0 + 1
{{1,2,3,4,5,6}}
=> [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
{{1,2,3,4,5},{6}}
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,3,4,6},{5}}
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
{{1,2,3,5,6},{4}}
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,3,5},{4,6}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
{{1,2,3,6},{4,5}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,3},{4,5,6}}
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,4,5,6},{3}}
=> [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,4,6},{3,5}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,4},{3,5,6}}
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2,4},{3},{5,6}}
=> [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
{{1,2,5,6},{3,4}}
=> [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,5},{3,4,6}}
=> [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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