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A259480
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T(n,m) counts connected skew Ferrers diagrams of shape lambda/mu with lambda and mu partitions of n and m respectively (0<=m<=n).
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11
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0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 5, 1, 0, 0, 0, 7, 2, 0, 0, 0, 0, 11, 5, 2, 0, 0, 0, 0, 15, 8, 4, 0, 0, 0, 0, 0, 22, 14, 10, 3, 0, 0, 0, 0, 0, 30, 21, 18, 7, 1, 0, 0, 0, 0, 0, 42, 32, 32, 17, 6, 0, 0, 0, 0, 0, 0, 56, 45, 50, 31, 15, 2, 0, 0, 0, 0, 0, 0, 77, 65, 80, 58, 36, 11, 2, 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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In contrast to A161492, which counts the same items by area and number of columns, this sequence appears to have no known generating function.
The diagonals T(n,n-k) count connected skew diagrams with weight k:
1; 2; 3,1; 5,2,2; 7,5,4,3,1; 11,8,10,7,6,2,2;
Their sums equal A006958.
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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(7,2) = 4, the pairs of partitions are ((4,3)/(2)), ((3,3,1)/(2), ((3,2,2)/(1,1)) and ((2,2,2,1)/(1,1));
the diagrams are:
x x 0 0 , x x 0 , x 0 0 , x 0
0 0 0 0 0 0 x 0 x 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; 5 1 0 0 0
n=5; 7 2 0 0 0 0
n=6; 11 5 2 0 0 0 0
n=7; 15 8 4 0 0 0 0 0
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MATHEMATICA
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(* see A259479 *) factor[\[Lambda]_, \[Mu]_]/; majorsweak[\[Lambda], \[Mu]]:=Block[{a1, a2, a3}, a1=Apply[Join, Table[{i, j}, {i, Length[\[Lambda]]}, {j, \[Lambda][[i]], \[Lambda][[Min[i+1, Length[\[Lambda]]]]], -1}]];
a2=Map[{First[#], First[#]>Length[\[Mu]]||\[Mu][[First[#]]]<#[[2]]}&, a1]; a3=Map[First, DeleteCases[SplitBy[a2, MatchQ[#, {_, False}]&], {{_, False}}], {2}];
Flatten[redu[Part[\[Lambda], #], Part[PadRight[\[Mu], Length[\[Lambda]], 0], #]/. 0->Sequence[]]&/@Map[Union, a3], 1]];
Table[Sum[Boole[majorsweak[\[Lambda], \[Mu]]&&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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Cf. A259478, A259479, A259481, A161492, A227309, A006958.
Sequence in context: A212209 A259481 A132825 * A280164 A049597 A210951
Adjacent sequences: A259477 A259478 A259479 * A259481 A259482 A259483
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KEYWORD
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nonn,tabl
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AUTHOR
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Wouter Meeussen, Jul 01 2015
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STATUS
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approved
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