Identifier
Values
([(0,3),(1,2)],4) => [1,1] => [1] => 0
([(1,4),(2,3)],5) => [1,1] => [1] => 0
([(0,1),(2,4),(3,4)],5) => [2,1] => [1] => 0
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1] => 0
([(2,5),(3,4)],6) => [1,1] => [1] => 0
([(1,2),(3,5),(4,5)],6) => [2,1] => [1] => 0
([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [1] => 0
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [2] => 0
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,1] => 0
([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [1] => 0
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1] => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [1] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1] => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [2] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [3] => 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => [1] => 0
([(3,6),(4,5)],7) => [1,1] => [1] => 0
([(2,3),(4,6),(5,6)],7) => [2,1] => [1] => 0
([(1,2),(3,6),(4,6),(5,6)],7) => [3,1] => [1] => 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [4,1] => [1] => 0
([(1,6),(2,6),(3,5),(4,5)],7) => [2,2] => [2] => 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [3,2] => [2] => 0
([(1,6),(2,5),(3,4)],7) => [1,1,1] => [1,1] => 0
([(1,2),(3,6),(4,5),(5,6)],7) => [3,1] => [1] => 0
([(0,3),(1,2),(4,6),(5,6)],7) => [2,1,1] => [1,1] => 0
([(2,3),(4,5),(4,6),(5,6)],7) => [3,1] => [1] => 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [4,1] => [1] => 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [4,1] => [1] => 0
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1] => 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => [4,1] => [1] => 0
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [3,2] => [2] => 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,2] => [2] => 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [5,1] => [1] => 0
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,2] => [2] => 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1] => 0
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1] => [1] => 0
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,3] => [3] => 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [6,1] => [1] => 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7,1] => [1] => 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => [4,2] => [2] => 0
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7) => [5,2] => [2] => 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7) => [4,1] => [1] => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,1,1] => [1,1] => 0
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [1] => 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [6,1] => [1] => 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5,1] => [1] => 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [3] => 0
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => [7,1] => [1] => 0
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3] => [3] => 0
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [4,3] => [3] => 0
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7,1] => [1] => 0
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [8,1] => [1] => 0
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => [7,1] => [1] => 0
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => [8,1] => [1] => 0
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [9,1] => [1] => 0
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) => [4,3] => [3] => 0
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,2] => [2] => 0
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3] => [3] => 0
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,3] => [3] => 0
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [10,1] => [1] => 0
search for individual values
searching the database for the individual values of this statistic
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:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Map
first row removal
Description
Removes the first entry of an integer partition
Map
to edge-partition of connected components
Description
Sends a graph to the partition recording the number of edges in its connected components.