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Identifier

St001804: Standard tableaux ⟶ ℤ

Values

[[1]] => 1

[[1,2]] => 1

[[1],[2]] => 1

[[1,2,3]] => 1

[[1,3],[2]] => 2

[[1,2],[3]] => 1

[[1],[2],[3]] => 1

[[1,2,3,4]] => 1

[[1,3,4],[2]] => 2

[[1,2,4],[3]] => 2

[[1,2,3],[4]] => 1

[[1,3],[2,4]] => 2

[[1,2],[3,4]] => 1

[[1,4],[2],[3]] => 3

[[1,3],[2],[4]] => 2

[[1,2],[3],[4]] => 2

[[1],[2],[3],[4]] => 1

[[1,2,3,4,5]] => 1

[[1,3,4,5],[2]] => 2

[[1,2,4,5],[3]] => 2

[[1,2,3,5],[4]] => 2

[[1,2,3,4],[5]] => 1

[[1,3,5],[2,4]] => 2

[[1,2,5],[3,4]] => 2

[[1,3,4],[2,5]] => 2

[[1,2,4],[3,5]] => 2

[[1,2,3],[4,5]] => 1

[[1,4,5],[2],[3]] => 3

[[1,3,5],[2],[4]] => 3

[[1,2,5],[3],[4]] => 3

[[1,3,4],[2],[5]] => 2

[[1,2,4],[3],[5]] => 2

[[1,2,3],[4],[5]] => 2

[[1,4],[2,5],[3]] => 3

[[1,3],[2,5],[4]] => 2

[[1,2],[3,5],[4]] => 1

[[1,3],[2,4],[5]] => 2

[[1,2],[3,4],[5]] => 1

[[1,5],[2],[3],[4]] => 4

[[1,4],[2],[3],[5]] => 3

[[1,3],[2],[4],[5]] => 3

[[1,2],[3],[4],[5]] => 3

[[1],[2],[3],[4],[5]] => 1

[[1,2,3,4,5,6]] => 1

[[1,3,4,5,6],[2]] => 2

[[1,2,4,5,6],[3]] => 2

[[1,2,3,5,6],[4]] => 2

[[1,2,3,4,6],[5]] => 2

[[1,2,3,4,5],[6]] => 1

[[1,3,5,6],[2,4]] => 2

[[1,2,5,6],[3,4]] => 2

[[1,3,4,6],[2,5]] => 2

[[1,2,4,6],[3,5]] => 2

[[1,2,3,6],[4,5]] => 2

[[1,3,4,5],[2,6]] => 2

[[1,2,4,5],[3,6]] => 2

[[1,2,3,5],[4,6]] => 2

[[1,2,3,4],[5,6]] => 1

[[1,4,5,6],[2],[3]] => 3

[[1,3,5,6],[2],[4]] => 3

[[1,2,5,6],[3],[4]] => 3

[[1,3,4,6],[2],[5]] => 3

[[1,2,4,6],[3],[5]] => 3

[[1,2,3,6],[4],[5]] => 3

[[1,3,4,5],[2],[6]] => 2

[[1,2,4,5],[3],[6]] => 2

[[1,2,3,5],[4],[6]] => 2

[[1,2,3,4],[5],[6]] => 2

[[1,3,5],[2,4,6]] => 2

[[1,2,5],[3,4,6]] => 2

[[1,3,4],[2,5,6]] => 2

[[1,2,4],[3,5,6]] => 2

[[1,2,3],[4,5,6]] => 1

[[1,4,6],[2,5],[3]] => 3

[[1,3,6],[2,5],[4]] => 3

[[1,2,6],[3,5],[4]] => 3

[[1,3,6],[2,4],[5]] => 3

[[1,2,6],[3,4],[5]] => 3

[[1,4,5],[2,6],[3]] => 3

[[1,3,5],[2,6],[4]] => 3

[[1,2,5],[3,6],[4]] => 3

[[1,3,4],[2,6],[5]] => 2

[[1,2,4],[3,6],[5]] => 2

[[1,2,3],[4,6],[5]] => 2

[[1,3,5],[2,4],[6]] => 2

[[1,2,5],[3,4],[6]] => 2

[[1,3,4],[2,5],[6]] => 2

[[1,2,4],[3,5],[6]] => 2

[[1,2,3],[4,5],[6]] => 2

[[1,5,6],[2],[3],[4]] => 4

[[1,4,6],[2],[3],[5]] => 4

[[1,3,6],[2],[4],[5]] => 4

[[1,2,6],[3],[4],[5]] => 4

[[1,4,5],[2],[3],[6]] => 3

[[1,3,5],[2],[4],[6]] => 3

[[1,2,5],[3],[4],[6]] => 3

[[1,3,4],[2],[5],[6]] => 3

[[1,2,4],[3],[5],[6]] => 3

[[1,2,3],[4],[5],[6]] => 3

[[1,4],[2,5],[3,6]] => 3

[[1,3],[2,5],[4,6]] => 2

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Description

The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.

Code

def statistic(T):
    if not T:
        return 0
    w = len(T[0])
    l = len(T)
    d = 0
    while d < l and len(T[d]) == w and T[0][w-1] < T[l-1-d][0]:
        d += 1
    return len(T)-d

Created

Jun 06, 2022 at 19:33 by Martin Rubey

Updated

Jun 06, 2022 at 19:33 by Martin Rubey