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Your data matches 26 different statistics following compositions of up to 3 maps.
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Matching statistic: St000708
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Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1]
=> 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1]
=> 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,-5,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,1,1]
=> 1
Description
The product of the parts of an integer partition.
Matching statistic: St001786
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001786: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001786: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,-1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,-4,1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-3,4,-1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,-3,-2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-4,3,2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,-5,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
Description
The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order.
Alternatively, remark that the monomials of the polynomial $\prod_{k=1}^n (z_1+\dots +z_k)$ are in bijection with Dyck paths, regarded as superdiagonal paths, with $n$ east steps: the exponent of $z_i$ is the number of north steps before the $i$-th east step, see [2]. Thus, this statistic records the coefficients of the monomials.
A formula for the coefficient of $z_1^{a_1}\dots z_n^{a_n}$ is provided in [3]:
$$
c_{(a_1,\dots,a_n)} = \prod_{k=1}^{n-1} \frac{n-k+1 - \sum_{i=k+1}^n a_i}{a_k!}.
$$
This polynomial arises in a partial symmetrization process as follows, see [1]. For $w\in\frak{S}_n$, let $w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$. Furthermore, let
$$G(\mathbf{x},\mathbf{z}) = \prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$$
Then $\sum_{w\in\frak{S}_{n+1}}w\cdot G = \prod_{k=1}^n (z_1+\dots +z_k)$.
Matching statistic: St000589
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,2,-4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,-3,-2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,-4,3,-2] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-2,-1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[2,1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-2,-1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-2,-1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-3,2,-1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[3,-4,1,-2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-3,4,-1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-3,-4,-1,-2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-4,2,3,-1] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[4,-3,-2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-4,3,2,-1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-4,-3,-2,-1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,2,3,4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,-3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,-2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[-1,2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,2,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,-5,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[5,6,7,8,3,4,1,2] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[7,8,5,6,3,4,1,2] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 8 - 1
[-4,3,6,-7,-8,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[5,6,3,4,-7,2,-8,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2 - 1
[4,-8,-6,-3,-7,1,2,5] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[-4,3,-8,-6,-7,1,2,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[-7,-6,5,8,3,1,2,4] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[5,-6,3,8,2,-7,1,4] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 2 - 1
[-7,-6,2,-5,3,4,-8,1] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[-5,2,6,8,3,-7,1,4] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 2 - 1
[5,6,7,1,8,2,3,4] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[8,-7,-6,3,1,2,4,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[2,-8,-7,-3,-6,1,4,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,4,7,8,1,2,5,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[3,7,2,6,-8,-5,1,4] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[5,8,-6,2,-7,1,3,4] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,-5,6,8,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,7,8,-6,4,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[7,8,4,5,-6,-3,1,2] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[2,7,8,-6,3,-5,1,4] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[2,7,8,-6,4,-5,1,3] => [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 6 - 1
[-6,4,7,8,3,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[1,2,5,6,7,8,3,4] => [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 1
[5,-4,3,-7,1,8,2,6] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 4 - 1
[8,7,1,5,-6,2,-4,3] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000609
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,2,-4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,-3,-2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,-4,3,-2] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-2,-1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[2,1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-2,-1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-2,-1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-3,2,-1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[3,-4,1,-2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-3,4,-1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-3,-4,-1,-2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-4,2,3,-1] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[4,-3,-2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-4,3,2,-1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-4,-3,-2,-1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,2,3,4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,-3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,-2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[-1,2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,2,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,-5,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[5,6,7,8,3,4,1,2] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[7,8,5,6,3,4,1,2] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 8 - 1
[-4,3,6,-7,-8,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[5,6,3,4,-7,2,-8,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2 - 1
[4,-8,-6,-3,-7,1,2,5] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[-4,3,-8,-6,-7,1,2,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[-7,-6,5,8,3,1,2,4] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[5,-6,3,8,2,-7,1,4] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 2 - 1
[-7,-6,2,-5,3,4,-8,1] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[-5,2,6,8,3,-7,1,4] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 2 - 1
[5,6,7,1,8,2,3,4] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[8,-7,-6,3,1,2,4,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[2,-8,-7,-3,-6,1,4,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,4,7,8,1,2,5,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[3,7,2,6,-8,-5,1,4] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[5,8,-6,2,-7,1,3,4] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,-5,6,8,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,7,8,-6,4,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[7,8,4,5,-6,-3,1,2] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[2,7,8,-6,3,-5,1,4] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[2,7,8,-6,4,-5,1,3] => [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 6 - 1
[-6,4,7,8,3,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[1,2,5,6,7,8,3,4] => [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 1
[5,-4,3,-7,1,8,2,6] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 4 - 1
[8,7,1,5,-6,2,-4,3] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000612
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,2,-4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,-3,-2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,-4,3,-2] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-2,-1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[2,1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-2,-1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-2,-1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-3,2,-1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[3,-4,1,-2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-3,4,-1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-3,-4,-1,-2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[-4,2,3,-1] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[4,-3,-2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-4,3,2,-1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[-4,-3,-2,-1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,2,3,4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,-3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,-3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,-2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,-2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[-1,2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,2,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,-5,4] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[5,6,7,8,3,4,1,2] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[7,8,5,6,3,4,1,2] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 8 - 1
[-4,3,6,-7,-8,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[5,6,3,4,-7,2,-8,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2 - 1
[4,-8,-6,-3,-7,1,2,5] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[-4,3,-8,-6,-7,1,2,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[-7,-6,5,8,3,1,2,4] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[5,-6,3,8,2,-7,1,4] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 2 - 1
[-7,-6,2,-5,3,4,-8,1] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[-5,2,6,8,3,-7,1,4] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 2 - 1
[5,6,7,1,8,2,3,4] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[8,-7,-6,3,1,2,4,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[2,-8,-7,-3,-6,1,4,5] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,4,7,8,1,2,5,6] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[3,7,2,6,-8,-5,1,4] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[5,8,-6,2,-7,1,3,4] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,-5,6,8,-4,1,2,7] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[3,7,8,-6,4,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[7,8,4,5,-6,-3,1,2] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 4 - 1
[2,7,8,-6,3,-5,1,4] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[2,7,8,-6,4,-5,1,3] => [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 6 - 1
[-6,4,7,8,3,-5,1,2] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
[1,2,5,6,7,8,3,4] => [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 1
[5,-4,3,-7,1,8,2,6] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 4 - 1
[8,7,1,5,-6,2,-4,3] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 3 - 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Matching statistic: St001232
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[-1,2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,-1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[-3,2,-1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,-4,1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-3,4,-1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[-4,2,3,-1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,-3,-2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-4,3,2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,5,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,3,-5,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,-3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,2,-3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,-2,3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,-2,3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[-1,2,3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[-1,2,3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,4,3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,2,4,-3,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,-4,3,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,2,-4,-3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,2,-4,-3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[-1,3,2,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,3,2,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,-2,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,-2,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,4,5,2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,4,-5,2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-4,5,-2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-4,-5,-2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,5,4,3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,5,-4,-3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-5,4,3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-5,-4,-3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,-3,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,-3,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,-1,-3,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,-1,-3,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,4,3,-5] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,-4,-3,-5] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,-1,4,3,-5] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,-1,-4,-3,-5] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,-5,-3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,-4,5,-3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,-4,-5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[-2,-1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[-2,-1,4,-5,-3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[-2,-1,-4,5,-3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[-2,-1,-4,-5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,5,-3,-4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,-5,3,-4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,-5,-3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[-2,-1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[-2,-1,5,-3,-4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[-2,-1,-5,3,-4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[-2,-1,-5,-3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,5,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,-5,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001593
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[1,2,3] => [-1,-2,-3] => [1,1,1]
=> 0 = 1 - 1
[1,2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,3,-4] => [-1,-2,-3,4] => [1,1,1]
=> 0 = 1 - 1
[1,2,-3,4] => [-1,-2,3,-4] => [1,1,1]
=> 0 = 1 - 1
[1,-2,3,4] => [-1,2,-3,-4] => [1,1,1]
=> 0 = 1 - 1
[-1,2,3,4] => [1,-2,-3,-4] => [1,1,1]
=> 0 = 1 - 1
[1,2,4,3] => [-1,-2,-4,-3] => [1,1]
=> 0 = 1 - 1
[1,2,-4,-3] => [-1,-2,4,3] => [1,1]
=> 0 = 1 - 1
[1,3,2,4] => [-1,-3,-2,-4] => [1,1]
=> 0 = 1 - 1
[1,-3,-2,4] => [-1,3,2,-4] => [1,1]
=> 0 = 1 - 1
[1,4,3,2] => [-1,-4,-3,-2] => [1,1]
=> 0 = 1 - 1
[1,-4,3,-2] => [-1,4,-3,2] => [1,1]
=> 0 = 1 - 1
[2,1,3,4] => [-2,-1,-3,-4] => [1,1]
=> 0 = 1 - 1
[-2,-1,3,4] => [2,1,-3,-4] => [1,1]
=> 0 = 1 - 1
[2,1,4,3] => [-2,-1,-4,-3] => []
=> ? = 2 - 1
[2,1,-4,-3] => [-2,-1,4,3] => []
=> ? = 2 - 1
[-2,-1,4,3] => [2,1,-4,-3] => []
=> ? = 2 - 1
[-2,-1,-4,-3] => [2,1,4,3] => []
=> ? = 2 - 1
[3,2,1,4] => [-3,-2,-1,-4] => [1,1]
=> 0 = 1 - 1
[-3,2,-1,4] => [3,-2,1,-4] => [1,1]
=> 0 = 1 - 1
[3,4,1,2] => [-3,-4,-1,-2] => []
=> ? = 2 - 1
[3,-4,1,-2] => [-3,4,-1,2] => []
=> ? = 2 - 1
[-3,4,-1,2] => [3,-4,1,-2] => []
=> ? = 2 - 1
[-3,-4,-1,-2] => [3,4,1,2] => []
=> ? = 2 - 1
[4,2,3,1] => [-4,-2,-3,-1] => [1,1]
=> 0 = 1 - 1
[-4,2,3,-1] => [4,-2,-3,1] => [1,1]
=> 0 = 1 - 1
[4,3,2,1] => [-4,-3,-2,-1] => []
=> ? = 2 - 1
[4,-3,-2,1] => [-4,3,2,-1] => []
=> ? = 2 - 1
[-4,3,2,-1] => [4,-3,-2,1] => []
=> ? = 2 - 1
[-4,-3,-2,-1] => [4,3,2,1] => []
=> ? = 2 - 1
[1,2,3,4,5] => [-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> 0 = 1 - 1
[1,2,3,4,-5] => [-1,-2,-3,-4,5] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,3,-4,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,3,-4,-5] => [-1,-2,-3,4,5] => [1,1,1]
=> 0 = 1 - 1
[1,2,-3,4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,-3,4,-5] => [-1,-2,3,-4,5] => [1,1,1]
=> 0 = 1 - 1
[1,2,-3,-4,5] => [-1,-2,3,4,-5] => [1,1,1]
=> 0 = 1 - 1
[1,-2,3,4,5] => [-1,2,-3,-4,-5] => [1,1,1,1]
=> 0 = 1 - 1
[1,-2,3,4,-5] => [-1,2,-3,-4,5] => [1,1,1]
=> 0 = 1 - 1
[1,-2,3,-4,5] => [-1,2,-3,4,-5] => [1,1,1]
=> 0 = 1 - 1
[1,-2,-3,4,5] => [-1,2,3,-4,-5] => [1,1,1]
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
=> 0 = 1 - 1
[-1,2,3,4,-5] => [1,-2,-3,-4,5] => [1,1,1]
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,-2,-3,4,-5] => [1,1,1]
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,-2,3,-4,-5] => [1,1,1]
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,2,-3,-4,-5] => [1,1,1]
=> 0 = 1 - 1
[1,2,3,5,4] => [-1,-2,-3,-5,-4] => [1,1,1]
=> 0 = 1 - 1
[1,2,3,5,-4] => [-1,-2,-3,-5,4] => [2,1,1,1]
=> 0 = 1 - 1
[1,2,3,-5,4] => [-1,-2,-3,5,-4] => [2,1,1,1]
=> 0 = 1 - 1
[1,2,3,-5,-4] => [-1,-2,-3,5,4] => [1,1,1]
=> 0 = 1 - 1
[1,2,-3,5,4] => [-1,-2,3,-5,-4] => [1,1]
=> 0 = 1 - 1
[1,2,-3,-5,-4] => [-1,-2,3,5,4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,5,4] => [-1,2,-3,-5,-4] => [1,1]
=> 0 = 1 - 1
[1,-2,3,-5,-4] => [-1,2,-3,5,4] => [1,1]
=> 0 = 1 - 1
[-1,2,3,5,4] => [1,-2,-3,-5,-4] => [1,1]
=> 0 = 1 - 1
[-1,2,3,-5,-4] => [1,-2,-3,5,4] => [1,1]
=> 0 = 1 - 1
[1,2,4,3,5] => [-1,-2,-4,-3,-5] => [1,1,1]
=> 0 = 1 - 1
[1,2,4,3,-5] => [-1,-2,-4,-3,5] => [1,1]
=> 0 = 1 - 1
[1,2,4,-3,5] => [-1,-2,-4,3,-5] => [2,1,1,1]
=> 0 = 1 - 1
[1,2,-4,3,5] => [-1,-2,4,-3,-5] => [2,1,1,1]
=> 0 = 1 - 1
[1,2,-4,-3,5] => [-1,-2,4,3,-5] => [1,1,1]
=> 0 = 1 - 1
[1,2,-4,-3,-5] => [-1,-2,4,3,5] => [1,1]
=> 0 = 1 - 1
[-1,3,2,5,4] => [1,-3,-2,-5,-4] => []
=> ? = 2 - 1
[-1,3,2,-5,-4] => [1,-3,-2,5,4] => []
=> ? = 2 - 1
[-1,-3,-2,5,4] => [1,3,2,-5,-4] => []
=> ? = 2 - 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => []
=> ? = 2 - 1
[-1,4,5,2,3] => [1,-4,-5,-2,-3] => []
=> ? = 2 - 1
[-1,4,-5,2,-3] => [1,-4,5,-2,3] => []
=> ? = 2 - 1
[-1,-4,5,-2,3] => [1,4,-5,2,-3] => []
=> ? = 2 - 1
[-1,-4,-5,-2,-3] => [1,4,5,2,3] => []
=> ? = 2 - 1
[-1,5,4,3,2] => [1,-5,-4,-3,-2] => []
=> ? = 2 - 1
[-1,5,-4,-3,2] => [1,-5,4,3,-2] => []
=> ? = 2 - 1
[-1,-5,4,3,-2] => [1,5,-4,-3,2] => []
=> ? = 2 - 1
[-1,-5,-4,-3,-2] => [1,5,4,3,2] => []
=> ? = 2 - 1
[2,1,-3,5,4] => [-2,-1,3,-5,-4] => []
=> ? = 2 - 1
[2,1,-3,-5,-4] => [-2,-1,3,5,4] => []
=> ? = 2 - 1
[-2,-1,-3,5,4] => [2,1,3,-5,-4] => []
=> ? = 2 - 1
[-2,-1,-3,-5,-4] => [2,1,3,5,4] => []
=> ? = 2 - 1
[2,1,4,3,-5] => [-2,-1,-4,-3,5] => []
=> ? = 2 - 1
[2,1,-4,-3,-5] => [-2,-1,4,3,5] => []
=> ? = 2 - 1
[-2,-1,4,3,-5] => [2,1,-4,-3,5] => []
=> ? = 2 - 1
[-2,-1,-4,-3,-5] => [2,1,4,3,5] => []
=> ? = 2 - 1
[2,1,5,-4,3] => [-2,-1,-5,4,-3] => []
=> ? = 2 - 1
[2,1,-5,-4,-3] => [-2,-1,5,4,3] => []
=> ? = 2 - 1
[-2,-1,5,-4,3] => [2,1,-5,4,-3] => []
=> ? = 2 - 1
[-2,-1,-5,-4,-3] => [2,1,5,4,3] => []
=> ? = 2 - 1
[3,-2,1,5,4] => [-3,2,-1,-5,-4] => []
=> ? = 2 - 1
[3,-2,1,-5,-4] => [-3,2,-1,5,4] => []
=> ? = 2 - 1
[-3,-2,-1,5,4] => [3,2,1,-5,-4] => []
=> ? = 2 - 1
[-3,-2,-1,-5,-4] => [3,2,1,5,4] => []
=> ? = 2 - 1
[3,4,1,2,-5] => [-3,-4,-1,-2,5] => []
=> ? = 2 - 1
[3,-4,1,-2,-5] => [-3,4,-1,2,5] => []
=> ? = 2 - 1
[-3,4,-1,2,-5] => [3,-4,1,-2,5] => []
=> ? = 2 - 1
[-3,-4,-1,-2,-5] => [3,4,1,2,5] => []
=> ? = 2 - 1
[3,5,1,-4,2] => [-3,-5,-1,4,-2] => []
=> ? = 2 - 1
[3,-5,1,-4,-2] => [-3,5,-1,4,2] => []
=> ? = 2 - 1
[-3,5,-1,-4,2] => [3,-5,1,4,-2] => []
=> ? = 2 - 1
[-3,-5,-1,-4,-2] => [3,5,1,4,2] => []
=> ? = 2 - 1
[4,-2,5,1,3] => [-4,2,-5,-1,-3] => []
=> ? = 2 - 1
[4,-2,-5,1,-3] => [-4,2,5,-1,3] => []
=> ? = 2 - 1
Description
This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
Matching statistic: St001283
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[1,2,3] => [-1,-2,-3] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4] => [-1,-2,-3,4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,4] => [-1,-2,3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,4] => [-1,2,-3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,4] => [1,-2,-3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,4,3] => [-1,-2,-4,-3] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,-4,-3] => [-1,-2,4,3] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,3,2,4] => [-1,-3,-2,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-3,-2,4] => [-1,3,2,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,4,3,2] => [-1,-4,-3,-2] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-4,3,-2] => [-1,4,-3,2] => [1,1]
=> [2]
=> 0 = 1 - 1
[2,1,3,4] => [-2,-1,-3,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-2,-1,3,4] => [2,1,-3,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[2,1,4,3] => [-2,-1,-4,-3] => []
=> []
=> ? = 2 - 1
[2,1,-4,-3] => [-2,-1,4,3] => []
=> []
=> ? = 2 - 1
[-2,-1,4,3] => [2,1,-4,-3] => []
=> []
=> ? = 2 - 1
[-2,-1,-4,-3] => [2,1,4,3] => []
=> []
=> ? = 2 - 1
[3,2,1,4] => [-3,-2,-1,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-3,2,-1,4] => [3,-2,1,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[3,4,1,2] => [-3,-4,-1,-2] => []
=> []
=> ? = 2 - 1
[3,-4,1,-2] => [-3,4,-1,2] => []
=> []
=> ? = 2 - 1
[-3,4,-1,2] => [3,-4,1,-2] => []
=> []
=> ? = 2 - 1
[-3,-4,-1,-2] => [3,4,1,2] => []
=> []
=> ? = 2 - 1
[4,2,3,1] => [-4,-2,-3,-1] => [1,1]
=> [2]
=> 0 = 1 - 1
[-4,2,3,-1] => [4,-2,-3,1] => [1,1]
=> [2]
=> 0 = 1 - 1
[4,3,2,1] => [-4,-3,-2,-1] => []
=> []
=> ? = 2 - 1
[4,-3,-2,1] => [-4,3,2,-1] => []
=> []
=> ? = 2 - 1
[-4,3,2,-1] => [4,-3,-2,1] => []
=> []
=> ? = 2 - 1
[-4,-3,-2,-1] => [4,3,2,1] => []
=> []
=> ? = 2 - 1
[1,2,3,4,5] => [-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
[1,2,3,4,-5] => [-1,-2,-3,-4,5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4,-5] => [-1,-2,-3,4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,-3,4,-5] => [-1,-2,3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,-4,5] => [-1,-2,3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,4,5] => [-1,2,-3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,-2,3,4,-5] => [-1,2,-3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,-4,5] => [-1,2,-3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,-3,4,5] => [-1,2,3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[-1,2,3,4,-5] => [1,-2,-3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,-2,-3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,-2,3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,2,-3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,5,4] => [-1,-2,-3,-5,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,5,-4] => [-1,-2,-3,-5,4] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,3,-5,4] => [-1,-2,-3,5,-4] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,3,-5,-4] => [-1,-2,-3,5,4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,5,4] => [-1,-2,3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,-3,-5,-4] => [-1,-2,3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-2,3,5,4] => [-1,2,-3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-2,3,-5,-4] => [-1,2,-3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-1,2,3,5,4] => [1,-2,-3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-1,2,3,-5,-4] => [1,-2,-3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,4,3,5] => [-1,-2,-4,-3,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,4,3,-5] => [-1,-2,-4,-3,5] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,4,-3,5] => [-1,-2,-4,3,-5] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,-4,3,5] => [-1,-2,4,-3,-5] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,-4,-3,5] => [-1,-2,4,3,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-4,-3,-5] => [-1,-2,4,3,5] => [1,1]
=> [2]
=> 0 = 1 - 1
[-1,3,2,5,4] => [1,-3,-2,-5,-4] => []
=> []
=> ? = 2 - 1
[-1,3,2,-5,-4] => [1,-3,-2,5,4] => []
=> []
=> ? = 2 - 1
[-1,-3,-2,5,4] => [1,3,2,-5,-4] => []
=> []
=> ? = 2 - 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => []
=> []
=> ? = 2 - 1
[-1,4,5,2,3] => [1,-4,-5,-2,-3] => []
=> []
=> ? = 2 - 1
[-1,4,-5,2,-3] => [1,-4,5,-2,3] => []
=> []
=> ? = 2 - 1
[-1,-4,5,-2,3] => [1,4,-5,2,-3] => []
=> []
=> ? = 2 - 1
[-1,-4,-5,-2,-3] => [1,4,5,2,3] => []
=> []
=> ? = 2 - 1
[-1,5,4,3,2] => [1,-5,-4,-3,-2] => []
=> []
=> ? = 2 - 1
[-1,5,-4,-3,2] => [1,-5,4,3,-2] => []
=> []
=> ? = 2 - 1
[-1,-5,4,3,-2] => [1,5,-4,-3,2] => []
=> []
=> ? = 2 - 1
[-1,-5,-4,-3,-2] => [1,5,4,3,2] => []
=> []
=> ? = 2 - 1
[2,1,-3,5,4] => [-2,-1,3,-5,-4] => []
=> []
=> ? = 2 - 1
[2,1,-3,-5,-4] => [-2,-1,3,5,4] => []
=> []
=> ? = 2 - 1
[-2,-1,-3,5,4] => [2,1,3,-5,-4] => []
=> []
=> ? = 2 - 1
[-2,-1,-3,-5,-4] => [2,1,3,5,4] => []
=> []
=> ? = 2 - 1
[2,1,4,3,-5] => [-2,-1,-4,-3,5] => []
=> []
=> ? = 2 - 1
[2,1,-4,-3,-5] => [-2,-1,4,3,5] => []
=> []
=> ? = 2 - 1
[-2,-1,4,3,-5] => [2,1,-4,-3,5] => []
=> []
=> ? = 2 - 1
[-2,-1,-4,-3,-5] => [2,1,4,3,5] => []
=> []
=> ? = 2 - 1
[2,1,5,-4,3] => [-2,-1,-5,4,-3] => []
=> []
=> ? = 2 - 1
[2,1,-5,-4,-3] => [-2,-1,5,4,3] => []
=> []
=> ? = 2 - 1
[-2,-1,5,-4,3] => [2,1,-5,4,-3] => []
=> []
=> ? = 2 - 1
[-2,-1,-5,-4,-3] => [2,1,5,4,3] => []
=> []
=> ? = 2 - 1
[3,-2,1,5,4] => [-3,2,-1,-5,-4] => []
=> []
=> ? = 2 - 1
[3,-2,1,-5,-4] => [-3,2,-1,5,4] => []
=> []
=> ? = 2 - 1
[-3,-2,-1,5,4] => [3,2,1,-5,-4] => []
=> []
=> ? = 2 - 1
[-3,-2,-1,-5,-4] => [3,2,1,5,4] => []
=> []
=> ? = 2 - 1
[3,4,1,2,-5] => [-3,-4,-1,-2,5] => []
=> []
=> ? = 2 - 1
[3,-4,1,-2,-5] => [-3,4,-1,2,5] => []
=> []
=> ? = 2 - 1
[-3,4,-1,2,-5] => [3,-4,1,-2,5] => []
=> []
=> ? = 2 - 1
[-3,-4,-1,-2,-5] => [3,4,1,2,5] => []
=> []
=> ? = 2 - 1
[3,5,1,-4,2] => [-3,-5,-1,4,-2] => []
=> []
=> ? = 2 - 1
[3,-5,1,-4,-2] => [-3,5,-1,4,2] => []
=> []
=> ? = 2 - 1
[-3,5,-1,-4,2] => [3,-5,1,4,-2] => []
=> []
=> ? = 2 - 1
[-3,-5,-1,-4,-2] => [3,5,1,4,2] => []
=> []
=> ? = 2 - 1
[4,-2,5,1,3] => [-4,2,-5,-1,-3] => []
=> []
=> ? = 2 - 1
[4,-2,-5,1,-3] => [-4,2,5,-1,3] => []
=> []
=> ? = 2 - 1
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers.
A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
Matching statistic: St001284
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 43%●distinct values known / distinct values provided: 33%
Values
[1,2,3] => [-1,-2,-3] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4] => [-1,-2,-3,4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,4] => [-1,-2,3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,4] => [-1,2,-3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,4] => [1,-2,-3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,4,3] => [-1,-2,-4,-3] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,-4,-3] => [-1,-2,4,3] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,3,2,4] => [-1,-3,-2,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-3,-2,4] => [-1,3,2,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,4,3,2] => [-1,-4,-3,-2] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-4,3,-2] => [-1,4,-3,2] => [1,1]
=> [2]
=> 0 = 1 - 1
[2,1,3,4] => [-2,-1,-3,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-2,-1,3,4] => [2,1,-3,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[2,1,4,3] => [-2,-1,-4,-3] => []
=> []
=> ? = 2 - 1
[2,1,-4,-3] => [-2,-1,4,3] => []
=> []
=> ? = 2 - 1
[-2,-1,4,3] => [2,1,-4,-3] => []
=> []
=> ? = 2 - 1
[-2,-1,-4,-3] => [2,1,4,3] => []
=> []
=> ? = 2 - 1
[3,2,1,4] => [-3,-2,-1,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-3,2,-1,4] => [3,-2,1,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[3,4,1,2] => [-3,-4,-1,-2] => []
=> []
=> ? = 2 - 1
[3,-4,1,-2] => [-3,4,-1,2] => []
=> []
=> ? = 2 - 1
[-3,4,-1,2] => [3,-4,1,-2] => []
=> []
=> ? = 2 - 1
[-3,-4,-1,-2] => [3,4,1,2] => []
=> []
=> ? = 2 - 1
[4,2,3,1] => [-4,-2,-3,-1] => [1,1]
=> [2]
=> 0 = 1 - 1
[-4,2,3,-1] => [4,-2,-3,1] => [1,1]
=> [2]
=> 0 = 1 - 1
[4,3,2,1] => [-4,-3,-2,-1] => []
=> []
=> ? = 2 - 1
[4,-3,-2,1] => [-4,3,2,-1] => []
=> []
=> ? = 2 - 1
[-4,3,2,-1] => [4,-3,-2,1] => []
=> []
=> ? = 2 - 1
[-4,-3,-2,-1] => [4,3,2,1] => []
=> []
=> ? = 2 - 1
[1,2,3,4,5] => [-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
[1,2,3,4,-5] => [-1,-2,-3,-4,5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4,-5] => [-1,-2,-3,4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,-3,4,-5] => [-1,-2,3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,-4,5] => [-1,-2,3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,4,5] => [-1,2,-3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,-2,3,4,-5] => [-1,2,-3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,-4,5] => [-1,2,-3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,-3,4,5] => [-1,2,3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[-1,2,3,4,-5] => [1,-2,-3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,-2,-3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,-2,3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,2,-3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,5,4] => [-1,-2,-3,-5,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,5,-4] => [-1,-2,-3,-5,4] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,3,-5,4] => [-1,-2,-3,5,-4] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,3,-5,-4] => [-1,-2,-3,5,4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,5,4] => [-1,-2,3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,-3,-5,-4] => [-1,-2,3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-2,3,5,4] => [-1,2,-3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-2,3,-5,-4] => [-1,2,-3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-1,2,3,5,4] => [1,-2,-3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-1,2,3,-5,-4] => [1,-2,-3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,4,3,5] => [-1,-2,-4,-3,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,4,3,-5] => [-1,-2,-4,-3,5] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,4,-3,5] => [-1,-2,-4,3,-5] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,-4,3,5] => [-1,-2,4,-3,-5] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,-4,-3,5] => [-1,-2,4,3,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-4,-3,-5] => [-1,-2,4,3,5] => [1,1]
=> [2]
=> 0 = 1 - 1
[-1,3,2,5,4] => [1,-3,-2,-5,-4] => []
=> []
=> ? = 2 - 1
[-1,3,2,-5,-4] => [1,-3,-2,5,4] => []
=> []
=> ? = 2 - 1
[-1,-3,-2,5,4] => [1,3,2,-5,-4] => []
=> []
=> ? = 2 - 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => []
=> []
=> ? = 2 - 1
[-1,4,5,2,3] => [1,-4,-5,-2,-3] => []
=> []
=> ? = 2 - 1
[-1,4,-5,2,-3] => [1,-4,5,-2,3] => []
=> []
=> ? = 2 - 1
[-1,-4,5,-2,3] => [1,4,-5,2,-3] => []
=> []
=> ? = 2 - 1
[-1,-4,-5,-2,-3] => [1,4,5,2,3] => []
=> []
=> ? = 2 - 1
[-1,5,4,3,2] => [1,-5,-4,-3,-2] => []
=> []
=> ? = 2 - 1
[-1,5,-4,-3,2] => [1,-5,4,3,-2] => []
=> []
=> ? = 2 - 1
[-1,-5,4,3,-2] => [1,5,-4,-3,2] => []
=> []
=> ? = 2 - 1
[-1,-5,-4,-3,-2] => [1,5,4,3,2] => []
=> []
=> ? = 2 - 1
[2,1,-3,5,4] => [-2,-1,3,-5,-4] => []
=> []
=> ? = 2 - 1
[2,1,-3,-5,-4] => [-2,-1,3,5,4] => []
=> []
=> ? = 2 - 1
[-2,-1,-3,5,4] => [2,1,3,-5,-4] => []
=> []
=> ? = 2 - 1
[-2,-1,-3,-5,-4] => [2,1,3,5,4] => []
=> []
=> ? = 2 - 1
[2,1,4,3,-5] => [-2,-1,-4,-3,5] => []
=> []
=> ? = 2 - 1
[2,1,-4,-3,-5] => [-2,-1,4,3,5] => []
=> []
=> ? = 2 - 1
[-2,-1,4,3,-5] => [2,1,-4,-3,5] => []
=> []
=> ? = 2 - 1
[-2,-1,-4,-3,-5] => [2,1,4,3,5] => []
=> []
=> ? = 2 - 1
[2,1,5,-4,3] => [-2,-1,-5,4,-3] => []
=> []
=> ? = 2 - 1
[2,1,-5,-4,-3] => [-2,-1,5,4,3] => []
=> []
=> ? = 2 - 1
[-2,-1,5,-4,3] => [2,1,-5,4,-3] => []
=> []
=> ? = 2 - 1
[-2,-1,-5,-4,-3] => [2,1,5,4,3] => []
=> []
=> ? = 2 - 1
[3,-2,1,5,4] => [-3,2,-1,-5,-4] => []
=> []
=> ? = 2 - 1
[3,-2,1,-5,-4] => [-3,2,-1,5,4] => []
=> []
=> ? = 2 - 1
[-3,-2,-1,5,4] => [3,2,1,-5,-4] => []
=> []
=> ? = 2 - 1
[-3,-2,-1,-5,-4] => [3,2,1,5,4] => []
=> []
=> ? = 2 - 1
[3,4,1,2,-5] => [-3,-4,-1,-2,5] => []
=> []
=> ? = 2 - 1
[3,-4,1,-2,-5] => [-3,4,-1,2,5] => []
=> []
=> ? = 2 - 1
[-3,4,-1,2,-5] => [3,-4,1,-2,5] => []
=> []
=> ? = 2 - 1
[-3,-4,-1,-2,-5] => [3,4,1,2,5] => []
=> []
=> ? = 2 - 1
[3,5,1,-4,2] => [-3,-5,-1,4,-2] => []
=> []
=> ? = 2 - 1
[3,-5,1,-4,-2] => [-3,5,-1,4,2] => []
=> []
=> ? = 2 - 1
[-3,5,-1,-4,2] => [3,-5,1,4,-2] => []
=> []
=> ? = 2 - 1
[-3,-5,-1,-4,-2] => [3,5,1,4,2] => []
=> []
=> ? = 2 - 1
[4,-2,5,1,3] => [-4,2,-5,-1,-3] => []
=> []
=> ? = 2 - 1
[4,-2,-5,1,-3] => [-4,2,5,-1,3] => []
=> []
=> ? = 2 - 1
Description
The number of finite groups that are realised by the given partition over the complex numbers.
A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
Matching statistic: St000929
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 37%●distinct values known / distinct values provided: 33%
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 37%●distinct values known / distinct values provided: 33%
Values
[1,2,3] => [-1,-2,-3] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4] => [-1,-2,-3,4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,4] => [-1,-2,3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,4] => [-1,2,-3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,4] => [1,-2,-3,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,4,3] => [-1,-2,-4,-3] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,-4,-3] => [-1,-2,4,3] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,3,2,4] => [-1,-3,-2,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-3,-2,4] => [-1,3,2,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,4,3,2] => [-1,-4,-3,-2] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-4,3,-2] => [-1,4,-3,2] => [1,1]
=> [2]
=> 0 = 1 - 1
[2,1,3,4] => [-2,-1,-3,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-2,-1,3,4] => [2,1,-3,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[2,1,4,3] => [-2,-1,-4,-3] => []
=> []
=> ? = 2 - 1
[2,1,-4,-3] => [-2,-1,4,3] => []
=> []
=> ? = 2 - 1
[-2,-1,4,3] => [2,1,-4,-3] => []
=> []
=> ? = 2 - 1
[-2,-1,-4,-3] => [2,1,4,3] => []
=> []
=> ? = 2 - 1
[3,2,1,4] => [-3,-2,-1,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-3,2,-1,4] => [3,-2,1,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[3,4,1,2] => [-3,-4,-1,-2] => []
=> []
=> ? = 2 - 1
[3,-4,1,-2] => [-3,4,-1,2] => []
=> []
=> ? = 2 - 1
[-3,4,-1,2] => [3,-4,1,-2] => []
=> []
=> ? = 2 - 1
[-3,-4,-1,-2] => [3,4,1,2] => []
=> []
=> ? = 2 - 1
[4,2,3,1] => [-4,-2,-3,-1] => [1,1]
=> [2]
=> 0 = 1 - 1
[-4,2,3,-1] => [4,-2,-3,1] => [1,1]
=> [2]
=> 0 = 1 - 1
[4,3,2,1] => [-4,-3,-2,-1] => []
=> []
=> ? = 2 - 1
[4,-3,-2,1] => [-4,3,2,-1] => []
=> []
=> ? = 2 - 1
[-4,3,2,-1] => [4,-3,-2,1] => []
=> []
=> ? = 2 - 1
[-4,-3,-2,-1] => [4,3,2,1] => []
=> []
=> ? = 2 - 1
[1,2,3,4,5] => [-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
[1,2,3,4,-5] => [-1,-2,-3,-4,5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,3,-4,-5] => [-1,-2,-3,4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,2,-3,4,-5] => [-1,-2,3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,-4,5] => [-1,-2,3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,4,5] => [-1,2,-3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[1,-2,3,4,-5] => [-1,2,-3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,3,-4,5] => [-1,2,-3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,-2,-3,4,5] => [-1,2,3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[-1,2,3,4,-5] => [1,-2,-3,-4,5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,-2,-3,4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,-2,3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,2,-3,-4,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,5,4] => [-1,-2,-3,-5,-4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,3,5,-4] => [-1,-2,-3,-5,4] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,3,-5,4] => [-1,-2,-3,5,-4] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,3,-5,-4] => [-1,-2,-3,5,4] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-3,5,4] => [-1,-2,3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,-3,-5,-4] => [-1,-2,3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-2,3,5,4] => [-1,2,-3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,-2,3,-5,-4] => [-1,2,-3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-1,2,3,5,4] => [1,-2,-3,-5,-4] => [1,1]
=> [2]
=> 0 = 1 - 1
[-1,2,3,-5,-4] => [1,-2,-3,5,4] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,4,3,5] => [-1,-2,-4,-3,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,4,3,-5] => [-1,-2,-4,-3,5] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,2,4,-3,5] => [-1,-2,-4,3,-5] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,-4,3,5] => [-1,-2,4,-3,-5] => [2,1,1,1]
=> [4,1]
=> 0 = 1 - 1
[1,2,-4,-3,5] => [-1,-2,4,3,-5] => [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,2,-4,-3,-5] => [-1,-2,4,3,5] => [1,1]
=> [2]
=> 0 = 1 - 1
[1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
=> [1]
=> ? = 2 - 1
[1,3,2,-5,-4] => [-1,-3,-2,5,4] => [1]
=> [1]
=> ? = 2 - 1
[1,-3,-2,5,4] => [-1,3,2,-5,-4] => [1]
=> [1]
=> ? = 2 - 1
[1,-3,-2,-5,-4] => [-1,3,2,5,4] => [1]
=> [1]
=> ? = 2 - 1
[-1,3,2,5,4] => [1,-3,-2,-5,-4] => []
=> []
=> ? = 2 - 1
[-1,3,2,-5,-4] => [1,-3,-2,5,4] => []
=> []
=> ? = 2 - 1
[-1,-3,-2,5,4] => [1,3,2,-5,-4] => []
=> []
=> ? = 2 - 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => []
=> []
=> ? = 2 - 1
[1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
=> [1]
=> ? = 2 - 1
[1,4,-5,2,-3] => [-1,-4,5,-2,3] => [1]
=> [1]
=> ? = 2 - 1
[1,-4,5,-2,3] => [-1,4,-5,2,-3] => [1]
=> [1]
=> ? = 2 - 1
[1,-4,-5,-2,-3] => [-1,4,5,2,3] => [1]
=> [1]
=> ? = 2 - 1
[-1,4,5,2,3] => [1,-4,-5,-2,-3] => []
=> []
=> ? = 2 - 1
[-1,4,-5,2,-3] => [1,-4,5,-2,3] => []
=> []
=> ? = 2 - 1
[-1,-4,5,-2,3] => [1,4,-5,2,-3] => []
=> []
=> ? = 2 - 1
[-1,-4,-5,-2,-3] => [1,4,5,2,3] => []
=> []
=> ? = 2 - 1
[1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
=> [1]
=> ? = 2 - 1
[1,5,-4,-3,2] => [-1,-5,4,3,-2] => [1]
=> [1]
=> ? = 2 - 1
[1,-5,4,3,-2] => [-1,5,-4,-3,2] => [1]
=> [1]
=> ? = 2 - 1
[1,-5,-4,-3,-2] => [-1,5,4,3,2] => [1]
=> [1]
=> ? = 2 - 1
[-1,5,4,3,2] => [1,-5,-4,-3,-2] => []
=> []
=> ? = 2 - 1
[-1,5,-4,-3,2] => [1,-5,4,3,-2] => []
=> []
=> ? = 2 - 1
[-1,-5,4,3,-2] => [1,5,-4,-3,2] => []
=> []
=> ? = 2 - 1
[-1,-5,-4,-3,-2] => [1,5,4,3,2] => []
=> []
=> ? = 2 - 1
[2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
=> [1]
=> ? = 2 - 1
[2,1,3,-5,-4] => [-2,-1,-3,5,4] => [1]
=> [1]
=> ? = 2 - 1
[2,1,-3,5,4] => [-2,-1,3,-5,-4] => []
=> []
=> ? = 2 - 1
[2,1,-3,-5,-4] => [-2,-1,3,5,4] => []
=> []
=> ? = 2 - 1
[-2,-1,3,5,4] => [2,1,-3,-5,-4] => [1]
=> [1]
=> ? = 2 - 1
[-2,-1,3,-5,-4] => [2,1,-3,5,4] => [1]
=> [1]
=> ? = 2 - 1
[-2,-1,-3,5,4] => [2,1,3,-5,-4] => []
=> []
=> ? = 2 - 1
[-2,-1,-3,-5,-4] => [2,1,3,5,4] => []
=> []
=> ? = 2 - 1
[2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
=> [1]
=> ? = 2 - 1
[2,1,4,3,-5] => [-2,-1,-4,-3,5] => []
=> []
=> ? = 2 - 1
[2,1,-4,-3,5] => [-2,-1,4,3,-5] => [1]
=> [1]
=> ? = 2 - 1
[2,1,-4,-3,-5] => [-2,-1,4,3,5] => []
=> []
=> ? = 2 - 1
[-2,-1,4,3,5] => [2,1,-4,-3,-5] => [1]
=> [1]
=> ? = 2 - 1
[-2,-1,4,3,-5] => [2,1,-4,-3,5] => []
=> []
=> ? = 2 - 1
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000933The number of multipartitions of sizes given by an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset.
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