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Your data matches 61 different statistics following compositions of up to 3 maps.
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Matching statistic: St001059
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St001059: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001059: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0
{{1,2}}
=> [2,1] => 0
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 0
{{1,2},{3}}
=> [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 0
Description
Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation.
Matching statistic: St000175
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1]
=> [1]
=> []
=> ? = 0
{{1,2}}
=> [2]
=> [2]
=> []
=> ? = 0
{{1},{2}}
=> [1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3}}
=> [3]
=> [2,1]
=> [1]
=> 0
{{1,2},{3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
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: St000749
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1]
=> [1]
=> []
=> ? = 0
{{1,2}}
=> [2]
=> [2]
=> []
=> ? = 0
{{1},{2}}
=> [1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3}}
=> [3]
=> [2,1]
=> [1]
=> 0
{{1,2},{3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,4,4}
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St001586
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1]
=> [1]
=> []
=> ? = 0
{{1,2}}
=> [2]
=> [2]
=> []
=> ? = 0
{{1},{2}}
=> [1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3}}
=> [3]
=> [2,1]
=> [1]
=> 0
{{1,2},{3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,5},{2,3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [3,2]
=> [2]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,2,4,4}
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St000205
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5},{2,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,6},{2,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5},{2,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,6},{2,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000225
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5},{2,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,6},{2,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000944
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5},{2,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,6},{2,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
This stupid comment should not be accepted as an edit!
Matching statistic: St001175
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5},{2,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,6},{2,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001178
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001178: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001178: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 77%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1]
=> []
=> ?
=> ? = 0
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,5},{2,6}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
{{1,3,4,6},{2,5}}
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,4,4}
Description
Twelve times the variance of the major index among all standard Young tableaux of a partition.
For a partition $\lambda$ of $n$, this variance is given in [1, Proposition 3.2] by
$$\frac{1}{12}\Big(\sum_{k = 1}^n i^2 - \sum_{i,j \in \lambda} h_{ij}^2\Big),$$
where the second sum ranges over all cells in $\lambda$ and $h_{ij}$ is the hook length of the cell $(i,j) \in \lambda$.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001570The minimal number of edges to add to make a graph Hamiltonian. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000666The number of right tethers of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001866The nesting alignments of a signed permutation. St001890The maximum magnitude of the Möbius function of a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001429The number of negative entries in a signed permutation. St000068The number of minimal elements in a poset. St001862The number of crossings of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001768The number of reduced words of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.
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