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Identifier

St000174: Semistandard tableaux ⟶ ℤ

Values

[[1]] => 0

[[2]] => 1

[[3]] => 1

[[4]] => 1

[[1,1]] => 0

[[1,2]] => 0

[[2,2]] => 1

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

[[1,3]] => 0

[[2,3]] => 1

[[3,3]] => 1

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

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

[[1,4]] => 0

[[2,4]] => 1

[[3,4]] => 1

[[4,4]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The flush statistic of a semistandard tableau.
Let $T$ be a tableaux with $r$ rows such that each row is longer than the row beneath it by at least one box. Let $1 \leq i < k \leq r+1$ and suppose $l$ is the smallest integer greater than $k$ such that there exists an $l$-segment in the $(i+1)$-st row of $T$. A $k$-segment in the $i$-th row of $T$ is called flush if the leftmost box in the $k$-segment and the leftmost box of the $l$-segment are in the same column of $T$. If, however, no such $l$ exists, then this $k$-segment is said to be flush if the number of boxes in the $k$-segment is equal to difference of the number of boxes between the $i$-th row and $(i+1)$-st row. The flush statistic is given by the number of $k$-segments in $T$.

References

[1] Salisbury, B. The flush statistic on semistandard Young tableaux arXiv:1401.1185

Code

def statistic(T):
    f = 0
    segments = {}
    for r in range(len(T)):
        for c in range(len(T[r])):
            for j in range(c+1):
                if T[r][j] != r+1:
                    if (r,T[r][j]) not in segments.keys():
                        segments[(r,T[r][j])] = j
    L = segments.items()
    for s in segments.iteritems():
        if s[0][0] != len(T)-1 and s[1] == len(T[s[0][0]+1]) and T[s[0][0]+1][-1] <= s[0][1]:
            f += 1
        if s[0][0] == len(T)-1 and s[1] == 0:
            f += 1
        else:
            for t in L:
                if s[0][0]+1 == t[0][0] and s[1] == t[1]:
                    if s[1] >= 1 and T[s[0][0]+1][s[1]] != T[s[0][0]+1][s[1]-1]:
                        f += 1
                    if s[1] < 1 and T[s[0][0]+1][s[1]] != s[0][0]+2:
                        f += 1
    return f

Created

Jan 15, 2014 at 19:13 by Ben Salisbury

Updated

Oct 19, 2015 at 16:34 by Christian Stump