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Identifier
Values
[] => 0
[1] => 1
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 2
[1,1,1] => 1
[4] => 1
[3,1] => 2
[2,2] => 2
[2,1,1] => 2
[1,1,1,1] => 1
[5] => 1
[4,1] => 2
[3,2] => 2
[3,1,1] => 2
[2,2,1] => 2
[2,1,1,1] => 2
[1,1,1,1,1] => 1
[6] => 1
[5,1] => 2
[4,2] => 2
[4,1,1] => 2
[3,3] => 2
[3,2,1] => 3
[3,1,1,1] => 2
[2,2,2] => 2
[2,2,1,1] => 2
[2,1,1,1,1] => 2
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 2
[5,2] => 2
[5,1,1] => 2
[4,3] => 2
[4,2,1] => 3
[4,1,1,1] => 2
[3,3,1] => 3
[3,2,2] => 3
[3,2,1,1] => 3
[3,1,1,1,1] => 2
[2,2,2,1] => 2
[2,2,1,1,1] => 2
[2,1,1,1,1,1] => 2
[1,1,1,1,1,1,1] => 1
[8] => 1
[7,1] => 2
[6,2] => 2
[6,1,1] => 2
[5,3] => 2
[5,2,1] => 3
[5,1,1,1] => 2
[4,4] => 2
[4,3,1] => 3
[4,2,2] => 3
[4,2,1,1] => 3
[4,1,1,1,1] => 2
[3,3,2] => 3
[3,3,1,1] => 3
[3,2,2,1] => 3
[3,2,1,1,1] => 3
[3,1,1,1,1,1] => 2
[2,2,2,2] => 2
[2,2,2,1,1] => 2
[2,2,1,1,1,1] => 2
[2,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1] => 1
[9] => 1
[8,1] => 2
[7,2] => 2
[7,1,1] => 2
[6,3] => 2
[6,2,1] => 3
[6,1,1,1] => 2
[5,4] => 2
[5,3,1] => 3
[5,2,2] => 3
[5,2,1,1] => 3
[5,1,1,1,1] => 2
[4,4,1] => 3
[4,3,2] => 3
[4,3,1,1] => 3
[4,2,2,1] => 3
[4,2,1,1,1] => 3
[4,1,1,1,1,1] => 2
[3,3,3] => 3
[3,3,2,1] => 3
[3,3,1,1,1] => 3
[3,2,2,2] => 3
[3,2,2,1,1] => 3
[3,2,1,1,1,1] => 3
[3,1,1,1,1,1,1] => 2
[2,2,2,2,1] => 2
[2,2,2,1,1,1] => 2
[2,2,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1] => 1
[10] => 1
[9,1] => 2
[8,2] => 2
[8,1,1] => 2
>>> Load all 228 entries. <<<
[7,3] => 2
[7,2,1] => 3
[7,1,1,1] => 2
[6,4] => 2
[6,3,1] => 3
[6,2,2] => 3
[6,2,1,1] => 3
[6,1,1,1,1] => 2
[5,5] => 2
[5,4,1] => 3
[5,3,2] => 3
[5,3,1,1] => 3
[5,2,2,1] => 3
[5,2,1,1,1] => 3
[5,1,1,1,1,1] => 2
[4,4,2] => 3
[4,4,1,1] => 3
[4,3,3] => 3
[4,3,2,1] => 4
[4,3,1,1,1] => 3
[4,2,2,2] => 3
[4,2,2,1,1] => 3
[4,2,1,1,1,1] => 3
[4,1,1,1,1,1,1] => 2
[3,3,3,1] => 3
[3,3,2,2] => 3
[3,3,2,1,1] => 3
[3,3,1,1,1,1] => 3
[3,2,2,2,1] => 3
[3,2,2,1,1,1] => 3
[3,2,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1] => 2
[2,2,2,2,2] => 2
[2,2,2,2,1,1] => 2
[2,2,2,1,1,1,1] => 2
[2,2,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1] => 1
[6,5] => 2
[5,5,1] => 3
[5,4,2] => 3
[5,4,1,1] => 3
[5,3,3] => 3
[5,3,2,1] => 4
[5,3,1,1,1] => 3
[5,2,2,2] => 3
[5,2,2,1,1] => 3
[4,4,3] => 3
[4,4,2,1] => 4
[4,4,1,1,1] => 3
[4,3,3,1] => 4
[4,3,2,2] => 4
[4,3,2,1,1] => 4
[4,2,2,2,1] => 3
[3,3,3,2] => 3
[3,3,3,1,1] => 3
[3,3,2,2,1] => 3
[3,2,2,2,2] => 3
[2,2,2,2,2,1] => 2
[6,6] => 2
[6,4,2] => 3
[5,5,2] => 3
[5,4,3] => 3
[5,4,2,1] => 4
[5,4,1,1,1] => 3
[5,3,3,1] => 4
[5,3,2,2] => 4
[5,3,2,1,1] => 4
[5,2,2,2,1] => 3
[4,4,4] => 3
[4,4,3,1] => 4
[4,4,2,2] => 4
[4,4,2,1,1] => 4
[4,3,3,2] => 4
[4,3,3,1,1] => 4
[4,3,2,2,1] => 4
[3,3,3,3] => 3
[3,3,3,2,1] => 3
[3,3,2,2,2] => 3
[3,3,2,2,1,1] => 3
[2,2,2,2,2,2] => 2
[5,5,3] => 3
[5,4,4] => 3
[5,4,3,1] => 4
[5,4,2,2] => 4
[5,4,2,1,1] => 4
[5,3,3,2] => 4
[5,3,3,1,1] => 4
[5,3,2,2,1] => 4
[4,4,4,1] => 4
[4,4,3,2] => 4
[4,4,3,1,1] => 4
[4,4,2,2,1] => 4
[4,3,3,3] => 4
[4,3,3,2,1] => 4
[3,3,3,3,1] => 3
[3,3,3,2,2] => 3
[5,5,4] => 3
[5,4,3,2] => 4
[5,4,3,1,1] => 4
[5,4,2,2,1] => 4
[5,3,3,2,1] => 4
[4,4,4,2] => 4
[4,4,3,3] => 4
[4,4,3,2,1] => 4
[3,3,3,3,2] => 3
[5,5,5] => 3
[5,4,3,2,1] => 5
[4,4,4,3] => 4
[3,3,3,3,3] => 3
[7,5,3,1] => 4
[4,4,4,4] => 4
[7,5,4,3,1] => 5
[6,5,4,3,2,1] => 6
[11,7,5,1] => 4
[9,7,5,3,1] => 5
[7,6,5,4,3,2,1] => 7
[9,7,5,5,3,1] => 6
[11,9,7,5,3,1] => 6
[11,8,7,5,4,1] => 6
[8,7,6,5,4,3,2,1] => 8
[11,9,7,6,5,3,1] => 7
[13,11,9,7,5,3,1] => 7
[13,11,9,7,7,5,3,1] => 8
[17,13,11,9,7,5,1] => 7
[15,13,11,9,7,5,3,1] => 8
[29,23,19,17,13,11,7,1] => 8
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Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
References
[1] Chow, T. Coloring a Ferrers diagram MathOverflow:203962
[2] wikipedia:Rook_polynomial
Code
def statistic(la):
    return min(p + i for i, p in enumerate(la + [0]))

def Ferrers_graph(mu):
    """Return the graph with vertices being the cells of the Ferrers
    diagram, two vertices are connected if the cells are in the same
    row or column.

    """
    V = mu.cells()
    G = Graph([V, lambda a,b: a[0] == b[0] or a[1] == b[1]], loops=False, multiedges=False)
    return G

@cached_function
def all_colouring_partitions(mu):
    if len(mu) > mu[0]:
        return all_colouring_partitions(mu.conjugate())
    
    from sage.graphs.graph_coloring import all_graph_colorings
    res = dict()
    for c in all_graph_colorings(Ferrers_graph(mu), max(mu[0], len(mu))):
        la = Partition(sorted((len(v) for v in c.values()), reverse=True))
        res[la] = res.get(la, 0) + 1
    return res

def statistic_alternative(mu):
    mu = Partition(mu)
    return max(la[0] for la in all_colouring_partitions(mu))
Created
Apr 19, 2017 at 10:21 by Martin Rubey
Updated
Dec 22, 2020 at 13:56 by Martin Rubey