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A259479
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Skew diagrams, both connected or not.
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10
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1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 3, 0, 0, 0, 7, 5, 2, 0, 0, 0, 11, 9, 6, 1, 0, 0, 0, 15, 13, 12, 6, 0, 0, 0, 0, 22, 20, 22, 14, 3, 0, 0, 0, 0, 30, 28, 36, 27, 13, 2, 0, 0, 0, 0, 42, 40, 56, 48, 31, 11, 1, 0, 0, 0, 0, 56, 54, 82, 77, 59, 33, 9, 0, 0, 0, 0, 0, 77, 75, 120, 121, 106, 72, 30, 6, 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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T(n,m) counts pairs of partitions lambda of n and mu of 0<=m<=n respectively, so that the Ferrers diagram of mu does not exceed that of lambda, and that the diagrams of lambda and mu do not contain equal rows or columns.
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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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Table of n, a(n) for n=0..90.
Wouter Meeussen, Table n, m, T(n,m) for n= 1..27
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EXAMPLE
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T(6,2) = 6, the pairs of partitions are ((4,2)/(2)), ((3,3)/(2), ((3,2,1)/(2)), ((3,2,1)/(1,1)), ((2,2,2)/(1,1)) and ((2,2,1,1)/(1,1))
and the diagrams are:
x x 0 0 , x x 0 , x x 0 , x 0 0 , x 0 , x 0
0 0 0 0 0 0 0 x 0 x 0 x 0
0 0 0 0 0
0
triangle begins:
k=0; 1 2 3 4 5 6
n=0; 1
n=1; 1 0
n=2; 2 0 0
n=3; 3 1 0 0
n=4; 5 3 0 0 0
n=5; 7 5 2 0 0 0
n=6; 11 9 6 1 0 0 0
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MATHEMATICA
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majorsweak[left_List, right_List]:=Block[{le1=Length[left], le2=Length[right]}, If[le2>le1||Min[Sign[left-PadRight[right, le1]]]<0, False, True]];
redu1[\[Lambda]_, \[Mu]_]/; majorsweak[\[Lambda], \[Mu]]:=Delete[#, List/@DeleteCases[Table[i Boole[\[Lambda][[i]]==\[Mu][[i]]], {i, Length[\[Mu]]}], 0]]&/@{\[Lambda], \[Mu]};
redu[\[Lambda]_, \[Mu]_]/; majorsweak[\[Lambda], \[Mu]]:=TransposePartition/@Apply[redu1, TransposePartition/@redu1[\[Lambda], \[Mu]]];
Table[Sum[Boole[majorsweak[\[Lambda], \[Mu]]&&redu[\[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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Cf. A259478, A259480, A259481, A161492, A227309, A006958.
Sequence in context: A288437 A287736 A180969 * A238343 A238128 A238121
Adjacent sequences: A259476 A259477 A259478 * A259480 A259481 A259482
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KEYWORD
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nonn,tabl
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AUTHOR
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Wouter Meeussen, Jun 28 2015
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STATUS
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approved
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