Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000175
St000175: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 2
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 0
[3,2,1]
=> 3
[3,1,1,1]
=> 3
[2,2,2]
=> 0
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 2
[3,2,2]
=> 2
[3,2,1,1]
=> 5
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 6
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 3
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial $$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$ The statistic of the degree of this polynomial. For example, the partition $(3, 2, 1, 1, 1)$ gives $$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$ which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$. This is the same as the number of unordered pairs of different parts, which follows from: $$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000766
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
St000766: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 0
[2]
=> [[1,2]]
=> [2] => 0
[1,1]
=> [[1],[2]]
=> [1,1] => 0
[3]
=> [[1,2,3]]
=> [3] => 0
[2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 0
[4]
=> [[1,2,3,4]]
=> [4] => 0
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => 0
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 0
[5]
=> [[1,2,3,4,5]]
=> [5] => 0
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [6] => 0
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 0
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 0
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => 0
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 3
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 5
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 6
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => 0
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => 3
Description
The number of inversions of an integer composition. This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.
Matching statistic: St000769
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
Mp00315: Integer compositions inverse Foata bijectionInteger compositions
St000769: Integer compositions ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 92%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [2] => [2] => 0
[1,1]
=> [[1],[2]]
=> [1,1] => [1,1] => 0
[3]
=> [[1,2,3]]
=> [3] => [3] => 0
[2,1]
=> [[1,2],[3]]
=> [2,1] => [2,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [1,1,1] => 0
[4]
=> [[1,2,3,4]]
=> [4] => [4] => 0
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [3,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2,2] => 0
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => 0
[5]
=> [[1,2,3,4,5]]
=> [5] => [5] => 0
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [4,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [3,2] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,3,1] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,2,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [6] => 0
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [5,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [4,2] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,4,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [3,3] => 0
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [3,2,1] => 3
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,1,3,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [2,2,2] => 0
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,1,2,1] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,1,1,2,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [7] => 0
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [6,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [5,2] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,5,1] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [4,3] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [4,2,1] => 3
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,1,4,1] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [3,3,1] => 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [2,3,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,3,2,1] => 5
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,1,1,3,1] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [2,2,2,1] => 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [1,2,1,2,1] => 6
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,1,1,1,2,1] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [8] => 0
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [7,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [6,2] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,6,1] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => [5,3] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [5,2,1] => 3
[4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [4,4,1,1] => [4,1,4,1] => ? = 4
[3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => [3,2,3,2] => ? = 4
[3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [3,3,1,1,1,1] => [1,1,3,1,3,1] => ? = 8
[2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [2,2,1,1,1,1,1,1] => [1,1,1,1,2,1,2,1] => ? = 12
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [6,6] => ? => ? = 0
[4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [4,4,2,2] => ? => ? = 4
[3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> [3,3,2,2,1,1] => ? => ? = 12
[2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [2,2,2,2,2,2] => ? => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [1,1,1,1,1,1,1,1,1,1,1,1] => ? => ? = 0
Description
The major index of a composition regarded as a word. This is the sum of the positions of the descents of the composition. For the statistic which interprets the composition as a descent set, see [[St000008]].
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000585: Set partitions ⟶ ℤResult quality: 43% values known / values provided: 43%distinct values known / distinct values provided: 58%
Values
[1]
=> [[1]]
=> {{1}}
=> ? = 0
[2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 3
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 4
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> {{1,2,5,6,7},{3,4}}
=> 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> 2
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> 2
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 5
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> 4
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 3
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> {{1,5},{2,7},{3},{4},{6}}
=> 6
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> {{1,7},{2},{3},{4},{5},{6}}
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> {{1,2,3,4,5,6,7,8}}
=> 0
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> {{1,3,4,5,6,7,8},{2}}
=> 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> {{1,2,5,6,7,8},{3,4}}
=> 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> {{1,4,5,6,7,8},{2},{3}}
=> 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 3
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 3
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 2
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 5
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 4
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> ? = 4
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 5
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> {{1,5,8},{2,7},{3},{4},{6}}
=> ? = 7
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 6
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> ? = 8
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 0
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? = 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> {{1,2,5,6,7,8,9},{3,4}}
=> ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> {{1,4,5,6,7,8,9},{2},{3}}
=> ? = 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> {{1,2,3,7,8,9},{4,5,6}}
=> ? = 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> {{1,3,6,7,8,9},{2,5},{4}}
=> ? = 3
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> {{1,5,6,7,8,9},{2},{3},{4}}
=> ? = 3
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> {{1,2,3,4,9},{5,6,7,8}}
=> ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> {{1,3,4,8,9},{2,6,7},{5}}
=> ? = 3
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> {{1,2,7,8,9},{3,4},{5,6}}
=> ? = 2
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> {{1,4,7,8,9},{2,6},{3},{5}}
=> ? = 5
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> {{1,6,7,8,9},{2},{3},{4},{5}}
=> ? = 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> {{1,3,4,5},{2,7,8,9},{6}}
=> ? = 2
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> {{1,2,5,9},{3,4,8},{6,7}}
=> ? = 3
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> {{1,4,5,9},{2,7,8},{3},{6}}
=> ? = 5
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> {{1,3,8,9},{2,5},{4,7},{6}}
=> ? = 5
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> {{1,5,8,9},{2,7},{3},{4},{6}}
=> ? = 7
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> {{1,7,8,9},{2},{3},{4},{5},{6}}
=> ? = 5
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? = 0
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> {{1,3,6},{2,5,9},{4,8},{7}}
=> ? = 5
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> {{1,5,6},{2,8,9},{3},{4},{7}}
=> ? = 6
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> {{1,2,9},{3,4},{5,6},{7,8}}
=> ? = 3
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> {{1,4,9},{2,6},{3,8},{5},{7}}
=> ? = 8
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> {{1,6,9},{2,8},{3},{4},{5},{7}}
=> ? = 9
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> {{1,3},{2,5},{4,7},{6,9},{8}}
=> ? = 4
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> {{1,5},{2,7},{3,9},{4},{6},{8}}
=> ? = 9
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> {{1,7},{2,9},{3},{4},{5},{6},{8}}
=> ? = 10
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 7
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 0
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> {{1,3,4,5,6,7,8,9,10},{2}}
=> ? = 1
Description
The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block.
Matching statistic: St001781
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St001781: Set partitions ⟶ ℤResult quality: 40% values known / values provided: 40%distinct values known / distinct values provided: 58%
Values
[1]
=> [[1]]
=> {{1}}
=> 0
[2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 3
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 4
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> {{1,2,5,6,7},{3,4}}
=> 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> 2
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> 2
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 5
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> 4
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 3
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> {{1,5},{2,7},{3},{4},{6}}
=> 6
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> {{1,7},{2},{3},{4},{5},{6}}
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> {{1,2,3,4,5,6,7,8}}
=> ? = 0
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> {{1,3,4,5,6,7,8},{2}}
=> ? = 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> {{1,2,5,6,7,8},{3,4}}
=> ? = 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> {{1,4,5,6,7,8},{2},{3}}
=> ? = 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 3
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 3
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 2
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 5
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 4
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> ? = 4
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 5
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> {{1,5,8},{2,7},{3},{4},{6}}
=> ? = 7
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 6
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> ? = 8
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 0
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? = 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> {{1,2,5,6,7,8,9},{3,4}}
=> ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> {{1,4,5,6,7,8,9},{2},{3}}
=> ? = 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> {{1,2,3,7,8,9},{4,5,6}}
=> ? = 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> {{1,3,6,7,8,9},{2,5},{4}}
=> ? = 3
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> {{1,5,6,7,8,9},{2},{3},{4}}
=> ? = 3
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> {{1,2,3,4,9},{5,6,7,8}}
=> ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> {{1,3,4,8,9},{2,6,7},{5}}
=> ? = 3
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> {{1,2,7,8,9},{3,4},{5,6}}
=> ? = 2
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> {{1,4,7,8,9},{2,6},{3},{5}}
=> ? = 5
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> {{1,6,7,8,9},{2},{3},{4},{5}}
=> ? = 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> {{1,3,4,5},{2,7,8,9},{6}}
=> ? = 2
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> {{1,2,5,9},{3,4,8},{6,7}}
=> ? = 3
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> {{1,4,5,9},{2,7,8},{3},{6}}
=> ? = 5
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> {{1,3,8,9},{2,5},{4,7},{6}}
=> ? = 5
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> {{1,5,8,9},{2,7},{3},{4},{6}}
=> ? = 7
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> {{1,7,8,9},{2},{3},{4},{5},{6}}
=> ? = 5
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? = 0
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> {{1,3,6},{2,5,9},{4,8},{7}}
=> ? = 5
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> {{1,5,6},{2,8,9},{3},{4},{7}}
=> ? = 6
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> {{1,2,9},{3,4},{5,6},{7,8}}
=> ? = 3
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> {{1,4,9},{2,6},{3,8},{5},{7}}
=> ? = 8
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> {{1,6,9},{2,8},{3},{4},{5},{7}}
=> ? = 9
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> {{1,3},{2,5},{4,7},{6,9},{8}}
=> ? = 4
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> {{1,5},{2,7},{3,9},{4},{6},{8}}
=> ? = 9
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> {{1,7},{2,9},{3},{4},{5},{6},{8}}
=> ? = 10
Description
The interlacing number of a set partition. Let $\pi$ be a set partition of $\{1,\dots,n\}$ with $k$ blocks. To each block of $\pi$ we add the element $\infty$, which is larger than $n$. Then, an interlacing of $\pi$ is a pair of blocks $B=(B_1 < \dots < B_b < B_{b+1} = \infty)$ and $C=(C_1 < \dots < C_c < C_{c+1} = \infty)$ together with an index $1\leq i\leq \min(b, c)$, such that $B_i < C_i < B_{i+1} < C_{i+1}$.
Matching statistic: St000455
Mp00095: Integer partitions to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000455: Graphs ⟶ ℤResult quality: 8% values known / values provided: 14%distinct values known / distinct values provided: 8%
Values
[1]
=> 10 => [1,2] => ([(1,2)],3)
=> 0
[2]
=> 100 => [1,3] => ([(2,3)],4)
=> 0
[1,1]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[3]
=> 1000 => [1,4] => ([(3,4)],5)
=> 0
[2,1]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,1]
=> 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[4]
=> 10000 => [1,5] => ([(4,5)],6)
=> 0
[3,1]
=> 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,2]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[2,1,1]
=> 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1]
=> 11110 => [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)
=> 0
[5]
=> 100000 => [1,6] => ([(5,6)],7)
=> 0
[4,1]
=> 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,2]
=> 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[3,1,1]
=> 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,2,1]
=> 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[6]
=> 1000000 => [1,7] => ([(6,7)],8)
=> ? = 0
[5,1]
=> 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[4,2]
=> 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[4,1,1]
=> 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[3,3]
=> 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[3,2,1]
=> 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,2,2]
=> 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[7]
=> 10000000 => [1,8] => ([(7,8)],9)
=> ? = 0
[6,1]
=> 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[5,2]
=> 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[5,1,1]
=> 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
[4,3]
=> 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[4,2,1]
=> 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[3,3,1]
=> 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,2,2]
=> 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[8]
=> 100000000 => [1,9] => ([(8,9)],10)
=> ? = 0
[7,1]
=> 100000010 => [1,7,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[6,2]
=> 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[6,1,1]
=> 100000110 => [1,6,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[5,3]
=> 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[5,2,1]
=> 10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
[5,1,1,1]
=> 100001110 => [1,5,1,1,2] => ([(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
[4,4]
=> 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[4,3,1]
=> 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[4,2,2]
=> 1001100 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[4,2,1,1]
=> 10010110 => [1,3,2,1,2] => ([(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[4,1,1,1,1]
=> 100011110 => [1,4,1,1,1,2] => ([(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 4
[3,3,2]
=> 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,3,1,1]
=> 1100110 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[3,2,2,1]
=> 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[3,2,1,1,1]
=> 10101110 => [1,2,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 7
[3,1,1,1,1,1]
=> 100111110 => [1,3,1,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[2,2,2,2]
=> 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[2,2,2,1,1]
=> 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[2,2,1,1,1,1]
=> 11011110 => [1,1,2,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 8
[3,3,3]
=> 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.