Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000319
Mp00287: Ordered set partitions to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00287: Ordered set partitions to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001330
Mp00287: Ordered set partitions to compositionInteger compositions
Mp00041: Integer compositions conjugateInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001330: Graphs ⟶ ℤResult quality: 25% values known / values provided: 43%distinct values known / distinct values provided: 25%
Values
[{1,2},{3}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[{1,3},{2}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[{2,3},{1}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[{1},{2,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1},{2,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1},{3,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2},{1,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2},{1,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3},{1,2},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{4},{1,2},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3},{1,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{4},{1,3},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2},{3,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3},{2,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{4},{2,3},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,2},{3},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,2},{4},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,3},{2},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,4},{2},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,3},{4},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,4},{3},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2,3},{1},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2,4},{1},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3,4},{1},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2,3},{4},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{2,4},{3},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{3,4},{2},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[{1,2},{3,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,3},{2,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,4},{2,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,3},{1,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,4},{1,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{3,4},{1,2}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,2,3},{4}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1,2,4},{3}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1,3,4},{2}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{2,3,4},{1}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1},{2},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{2},{3,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{2},{4,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{3},{2,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{3},{2,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{4},{2,3},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{5},{2,3},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{4},{2,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{5},{2,4},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{3},{4,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{4},{3,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{5},{3,4},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2},{1},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2},{1},{3,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2},{1},{4,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3},{1},{2,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3},{1},{2,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4},{1},{2,3},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{5},{1},{2,3},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4},{1},{2,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{5},{1},{2,4},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3},{1},{4,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4},{1},{3,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1},{2,3},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{2,4},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{2,5},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{3,4},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{3,5},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{4,5},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{1,3},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{1,4},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{1,5},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{1,2},{4,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{1,2},{3,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{1,2},{3,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{1,4},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{1,5},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{1,3},{2,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{1,3},{2,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{1,5},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{1,4},{2,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{3,4},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{3,5},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{4,5},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{2,4},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{2,5},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{2,3},{1,5}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{2,3},{1,4}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{2,5},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{2,4},{1,3}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{3},{4,5},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{4},{3,5},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{5},{3,4},{1,2}] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{2,3,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{1},{2,3,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{1},{2,4,5},{3}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{1},{3,4,5},{2}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{2},{1,3,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{2},{1,3,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{2},{1,4,5},{3}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{3},{1,2,4},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{3},{1,2,5},{4}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[{4},{1,2,3},{5}] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
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.
Matching statistic: St000454
Mp00287: Ordered set partitions to compositionInteger compositions
Mp00041: Integer compositions conjugateInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000454: Graphs ⟶ ℤResult quality: 15% values known / values provided: 15%distinct values known / distinct values provided: 50%
Values
[{1,2},{3}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 2
[{1,3},{2}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 2
[{2,3},{1}] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 2
[{1},{2,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{1},{2,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{1},{3,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{2},{1,3},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{2},{1,4},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{3},{1,2},{4}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{4},{1,2},{3}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{3},{1,4},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{4},{1,3},{2}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{2},{3,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{3},{2,4},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{4},{2,3},{1}] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,2},{3},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,2},{4},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,3},{2},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,4},{2},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,3},{4},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,4},{3},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,3},{1},{4}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,4},{1},{3}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{3,4},{1},{2}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,3},{4},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,4},{3},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{3,4},{2},{1}] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,2},{3,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,3},{2,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,4},{2,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,3},{1,4}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{2,4},{1,3}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{3,4},{1,2}] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[{1,2,3},{4}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1,2,4},{3}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1,3,4},{2}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{2,3,4},{1}] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[{1},{2},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{2},{3,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{2},{4,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{3},{2,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{3},{2,5},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{4},{2,3},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{5},{2,3},{4}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{4},{2,5},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{5},{2,4},{3}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{3},{4,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{4},{3,5},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1},{5},{3,4},{2}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{2},{1},{3,4},{5}] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[{1,2},{3},{4},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,2},{3},{5},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,2},{4},{3},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,2},{5},{3},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,2},{4},{5},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,2},{5},{4},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,3},{2},{4},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,3},{2},{5},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,4},{2},{3},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,5},{2},{3},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,4},{2},{5},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,5},{2},{4},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,3},{4},{2},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,3},{5},{2},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,4},{3},{2},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,5},{3},{2},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,4},{5},{2},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,5},{4},{2},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,3},{4},{5},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,3},{5},{4},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,4},{3},{5},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,5},{3},{4},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,4},{5},{3},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{1,5},{4},{3},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,3},{1},{4},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,3},{1},{5},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,4},{1},{3},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,5},{1},{3},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,4},{1},{5},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,5},{1},{4},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3,4},{1},{2},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3,5},{1},{2},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4,5},{1},{2},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3,4},{1},{5},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3,5},{1},{4},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4,5},{1},{3},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,3},{4},{1},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,3},{5},{1},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,4},{3},{1},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,5},{3},{1},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,4},{5},{1},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,5},{4},{1},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3,4},{2},{1},{5}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3,5},{2},{1},{4}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4,5},{2},{1},{3}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3,4},{5},{1},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{3,5},{4},{1},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{4,5},{3},{1},{2}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,3},{4},{5},{1}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[{2,3},{5},{4},{1}] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Mp00287: Ordered set partitions to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
St001435: Skew partitions ⟶ ℤResult quality: 10% values known / values provided: 10%distinct values known / distinct values provided: 75%
Values
[{1,2},{3}] => [2,1] => [[2,2],[1]]
=> 1 = 0 + 1
[{1,3},{2}] => [2,1] => [[2,2],[1]]
=> 1 = 0 + 1
[{2,3},{1}] => [2,1] => [[2,2],[1]]
=> 1 = 0 + 1
[{1},{2,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{1},{2,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{1},{3,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{2},{1,3},{4}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{2},{1,4},{3}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{3},{1,2},{4}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{4},{1,2},{3}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{3},{1,4},{2}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{4},{1,3},{2}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{2},{3,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{3},{2,4},{1}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{4},{2,3},{1}] => [1,2,1] => [[2,2,1],[1]]
=> 1 = 0 + 1
[{1,2},{3},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{1,2},{4},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{1,3},{2},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{1,4},{2},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{1,3},{4},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{1,4},{3},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{2,3},{1},{4}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{2,4},{1},{3}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{3,4},{1},{2}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{2,3},{4},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{2,4},{3},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{3,4},{2},{1}] => [2,1,1] => [[2,2,2],[1,1]]
=> 1 = 0 + 1
[{1,2},{3,4}] => [2,2] => [[3,2],[1]]
=> 1 = 0 + 1
[{1,3},{2,4}] => [2,2] => [[3,2],[1]]
=> 1 = 0 + 1
[{1,4},{2,3}] => [2,2] => [[3,2],[1]]
=> 1 = 0 + 1
[{2,3},{1,4}] => [2,2] => [[3,2],[1]]
=> 1 = 0 + 1
[{2,4},{1,3}] => [2,2] => [[3,2],[1]]
=> 1 = 0 + 1
[{3,4},{1,2}] => [2,2] => [[3,2],[1]]
=> 1 = 0 + 1
[{1,2,3},{4}] => [3,1] => [[3,3],[2]]
=> 2 = 1 + 1
[{1,2,4},{3}] => [3,1] => [[3,3],[2]]
=> 2 = 1 + 1
[{1,3,4},{2}] => [3,1] => [[3,3],[2]]
=> 2 = 1 + 1
[{2,3,4},{1}] => [3,1] => [[3,3],[2]]
=> 2 = 1 + 1
[{1},{2},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{2},{3,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{2},{4,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{3},{2,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{3},{2,5},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{4},{2,3},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{5},{2,3},{4}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{4},{2,5},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{5},{2,4},{3}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{3},{4,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{4},{3,5},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{5},{3,4},{2}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{2},{1},{3,4},{5}] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1 = 0 + 1
[{1},{2},{3},{4,5},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{3},{4,6},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{3},{5,6},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{4},{3,5},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{4},{3,6},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{5},{3,4},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{6},{3,4},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{5},{3,6},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{6},{3,5},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{4},{5,6},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{5},{4,6},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{2},{6},{4,5},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{2},{4,5},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{2},{4,6},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{2},{5,6},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{2},{3,5},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{2},{3,6},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{2},{3,4},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{2},{3,4},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{2},{3,6},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{2},{3,5},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{2},{5,6},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{2},{4,6},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{2},{4,5},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{4},{2,5},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{4},{2,6},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{5},{2,4},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{6},{2,4},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{5},{2,6},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{6},{2,5},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{3},{2,5},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{3},{2,6},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{3},{2,4},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{3},{2,4},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{3},{2,6},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{3},{2,5},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{5},{2,3},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{6},{2,3},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{4},{2,3},{6}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{4},{2,3},{5}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{6},{2,3},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{5},{2,3},{4}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{5},{2,6},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{4},{6},{2,5},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{4},{2,6},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{4},{2,5},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{5},{6},{2,4},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{6},{5},{2,4},{3}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{4},{5,6},{2}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
[{1},{3},{5},{4,6},{2}] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 0 + 1
Description
The number of missing boxes in the first row.
Mp00287: Ordered set partitions to compositionInteger compositions
Mp00038: Integer compositions reverseInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001645: Graphs ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 25%
Values
[{1,2},{3}] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 6
[{1,3},{2}] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 6
[{2,3},{1}] => [2,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 6
[{1},{2,3},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1},{2,4},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1},{3,4},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2},{1,3},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2},{1,4},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3},{1,2},{4}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{4},{1,2},{3}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3},{1,4},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{4},{1,3},{2}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2},{3,4},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3},{2,4},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{4},{2,3},{1}] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,2},{3},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,2},{4},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,3},{2},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,4},{2},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,3},{4},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,4},{3},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,3},{1},{4}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,4},{1},{3}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3,4},{1},{2}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,3},{4},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,4},{3},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{3,4},{2},{1}] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,2},{3,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,3},{2,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,4},{2,3}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,3},{1,4}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{2,4},{1,3}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{3,4},{1,2}] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 6
[{1,2,3},{4}] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[{1,2,4},{3}] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[{1,3,4},{2}] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[{2,3,4},{1}] => [3,1] => [1,3] => ([(2,3)],4)
=> ? = 1 + 6
[{1},{2},{3,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{2},{3,5},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{2},{4,5},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{3},{2,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{3},{2,5},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{4},{2,3},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{5},{2,3},{4}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{4},{2,5},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{5},{2,4},{3}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{3},{4,5},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{4},{3,5},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{5},{3,4},{2}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{2},{1},{3,4},{5}] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 6
[{1},{2},{3},{4,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{3},{4,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{3},{5,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{4},{3,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{4},{3,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{5},{3,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{6},{3,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{5},{3,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{6},{3,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{4},{5,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{5},{4,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{2},{6},{4,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{2},{4,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{2},{4,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{2},{5,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{2},{3,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{2},{3,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{2},{3,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{2},{3,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{2},{3,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{2},{3,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{2},{5,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{2},{4,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{2},{4,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{4},{2,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{4},{2,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{5},{2,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{6},{2,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{5},{2,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{6},{2,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{3},{2,5},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{3},{2,6},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{3},{2,4},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{3},{2,4},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{3},{2,6},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{3},{2,5},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{5},{2,3},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{6},{2,3},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{4},{2,3},{6}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{4},{2,3},{5}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{6},{2,3},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{5},{2,3},{4}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{5},{2,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{4},{6},{2,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{4},{2,6},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{4},{2,5},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{5},{6},{2,4},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{6},{5},{2,4},{3}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{4},{5,6},{2}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
[{1},{3},{5},{4,6},{2}] => [1,1,1,2,1] => [1,2,1,1,1] => ([(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 = 0 + 6
Description
The pebbling number of a connected graph.