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A259481
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T(n,m) counts of border strips in skew tabloids of shape lambda/mu, with lambda and mu partitions of n and m (0<=m<=n).
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4
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0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 5, 2, 0, 0, 0, 0, 6, 3, 2, 0, 0, 0, 0, 7, 4, 4, 0, 0, 0, 0, 0, 8, 5, 6, 3, 0, 0, 0, 0, 0, 9, 6, 8, 6, 1, 0, 0, 0, 0, 0, 10, 7, 10, 9, 6, 0, 0, 0, 0, 0, 0, 11, 8, 12, 12, 11, 2, 0, 0, 0, 0, 0, 0, 12, 9, 14, 15, 16, 9, 2, 0, 0, 0, 0, 0, 0, 13, 10, 16, 18, 21, 16, 7, 0, 0, 0, 0, 0, 0, 0, 14, 11, 18, 21, 26, 23, 18, 4, 0, 0, 0, 0, 0, 0, 0, 15, 12, 20, 24, 31, 30, 29, 12, 3, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,4
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COMMENTS
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Border strips are defined as connected skew tabloids free of 2-by-2 cells.
Row sums are the partition numbers (A000041), diagonals sum to 2^n (A000079).
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REFERENCES
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I. G. MacDonald: "Symmetric functions and Hall polynomials"; Oxford University Press, 1979. Page 4.
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LINKS
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EXAMPLE
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T(8,2) = 6, the pairs of partitions are ((5,3)/(2)), ((4,3,1)/(2), ((4,2,2)/(1,1)), ((3,3,1,1)/(2)),
((3,2,2,1)/(1,1)) and ((2,2,2,1,1)/(1,1)); the diagrams are:
x x 0 0 0 , x x 0 0 , x 0 0 0 , x x 0 , x 0 0 , x 0
0 0 0 0 0 0 x 0 0 0 0 x 0 x 0
0 0 0 0 0 0 0 0
0 0 0
0
triangle begins:
k=0; 1 2 3 4 5 6 7
n=0; 0
n=1; 1 0
n=2; 2 0 0
n=3; 3 0 0 0
n=4; 4 1 0 0 0
n=5; 5 2 0 0 0 0
n=6; 6 3 2 0 0 0 0
n=7; 7 4 4 0 0 0 0 0
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MATHEMATICA
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(* see A259479 *) Table[Sum[Boole[majorsweak[\[Lambda], \[Mu]]&&( Tr[\[Lambda]]-Tr[\[Mu]]==Length[\[Lambda]]+First[\[Lambda]]-1 )&& redu[\[Lambda], \[Mu]]==factor[\[Lambda], \[Mu]]=={\[Lambda], \[Mu]}], {\[Lambda], Partitions[n]}, {\[Mu], Partitions[k]}], {n, 0, 12}, {k, 0, n}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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