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Identifier

St000176: Gelfand-Tsetlin patterns ⟶ ℤ

Values

[[1]] => 1

[[2]] => 1

[[3]] => 1

[[4]] => 1

[[5]] => 1

[[1,0],[0]] => 2

[[1,0],[1]] => 2

[[2,0],[0]] => 2

[[2,0],[1]] => 3

[[2,0],[2]] => 2

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

[[3,0],[0]] => 2

[[3,0],[1]] => 3

[[3,0],[2]] => 3

[[3,0],[3]] => 2

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

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

[[4,0],[0]] => 2

[[4,0],[1]] => 3

[[4,0],[2]] => 3

[[4,0],[3]] => 3

[[4,0],[4]] => 2

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

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

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

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

[[5,0],[0]] => 2

[[5,0],[1]] => 3

[[5,0],[2]] => 3

[[5,0],[3]] => 3

[[5,0],[4]] => 3

[[5,0],[5]] => 2

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

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

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

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

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

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

[[6,0],[0]] => 2

[[6,0],[1]] => 3

[[6,0],[2]] => 3

[[6,0],[3]] => 3

[[6,0],[4]] => 3

[[6,0],[5]] => 3

[[6,0],[6]] => 2

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

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

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

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

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

[[4,2],[2]] => 2

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

[[4,2],[4]] => 2

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

[[1,0,0],[0,0],[0]] => 2

[[1,0,0],[1,0],[0]] => 2

[[1,0,0],[1,0],[1]] => 2

[[2,0,0],[0,0],[0]] => 2

[[2,0,0],[1,0],[0]] => 3

[[2,0,0],[1,0],[1]] => 3

[[2,0,0],[2,0],[0]] => 2

[[2,0,0],[2,0],[1]] => 3

[[2,0,0],[2,0],[2]] => 2

[[1,1,0],[1,0],[0]] => 2

[[1,1,0],[1,0],[1]] => 2

[[1,1,0],[1,1],[1]] => 2

[[3,0,0],[0,0],[0]] => 2

[[3,0,0],[1,0],[0]] => 3

[[3,0,0],[1,0],[1]] => 3

[[3,0,0],[2,0],[0]] => 3

[[3,0,0],[2,0],[1]] => 4

[[3,0,0],[2,0],[2]] => 3

[[3,0,0],[3,0],[0]] => 2

[[3,0,0],[3,0],[1]] => 3

[[3,0,0],[3,0],[2]] => 3

[[3,0,0],[3,0],[3]] => 2

[[2,1,0],[1,0],[0]] => 3

[[2,1,0],[1,0],[1]] => 3

[[2,1,0],[1,1],[1]] => 3

[[2,1,0],[2,0],[0]] => 3

[[2,1,0],[2,0],[1]] => 4

[[2,1,0],[2,0],[2]] => 3

[[2,1,0],[2,1],[1]] => 3

[[2,1,0],[2,1],[2]] => 3

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

[[4,0,0],[0,0],[0]] => 2

[[4,0,0],[1,0],[0]] => 3

[[4,0,0],[1,0],[1]] => 3

[[4,0,0],[2,0],[0]] => 3

[[4,0,0],[2,0],[1]] => 4

[[4,0,0],[2,0],[2]] => 3

[[4,0,0],[3,0],[0]] => 3

[[4,0,0],[3,0],[1]] => 4

[[4,0,0],[3,0],[2]] => 4

[[4,0,0],[3,0],[3]] => 3

[[4,0,0],[4,0],[0]] => 2

[[4,0,0],[4,0],[1]] => 3

[[4,0,0],[4,0],[2]] => 3

[[4,0,0],[4,0],[3]] => 3

[[4,0,0],[4,0],[4]] => 2

[[3,1,0],[1,0],[0]] => 3

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Description

The total number of tiles in the Gelfand-Tsetlin pattern.
The tiling of a Gelfand-Tsetlin pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called tiles, and each entry in a pattern belongs to exactly one tile.

References

[1] De Loera, Jesús A., McAllister, T. B. Vertices of Gelfand-Tsetlin polytopes MathSciNet:2096742

Code

def statistic(P):
    """
    sage: P = GelfandTsetlinPattern([[5,4,2,1,0],[5,3,2,0],[5,3,2],[5,2],[4]])
    sage: statistic(P)
    7
    """
    places = [(i, j) for i in range(len(P)) for j in range(len(P[i]))]
    tiles = DisjointSet(places)
    n = len(P)
    for i, j in places:
        k, l = (i+1, j-1)
        if 0 <= k < n and 0 <= l < len(P[k]) and P[i][j] == P[k][l]:
            tiles.union((i,j), (k,l))
        k, l = (i+1, j)
        if 0 <= k < n and 0 <= l < len(P[k]) and P[i][j] == P[k][l]:
            tiles.union((i,j), (k,l))
        k, l = (i-1, j+1)
        if 0 <= k < n and 0 <= l < len(P[k]) and P[i][j] == P[k][l]:
            tiles.union((i,j), (k,l))
    return len(list(tiles))

Created

Mar 21, 2014 at 22:47 by Per Alexandersson

Updated

Jun 01, 2019 at 08:43 by Martin Rubey