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%I #22 Aug 28 2015 13:53:57
%S 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,
%T 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,
%U 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
%N 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).
%C In contrast to A161492, which counts the same items by area and number of columns, this sequence appears to have no known generating function.
%C The diagonals T(n,n-k) count connected skew diagrams with weight k:
%C 1; 2; 3,1; 5,2,2; 7,5,4,3,1; 11,8,10,7,6,2,2;
%C Their sums equal A006958.
%D I. G. MacDonald: "Symmetric functions and Hall polynomials"; Oxford University Press, 1979. Page 4.
%H Wouter Meeussen, <a href="/A259480/a259480.txt">Table n,m, T(n,m) for n= 1..27</a>
%e 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));
%e the diagrams are:
%e x x 0 0 , x x 0 , x 0 0 , x 0
%e 0 0 0 0 0 0 x 0 x 0
%e 0 0 0 0 0
%e 0
%e triangle begins:
%e k=0; 1 2 3 4 5 6 7
%e n=0; 0
%e n=1; 1 0
%e n=2; 2 0 0
%e n=3; 3 0 0 0
%e n=4; 5 1 0 0 0
%e n=5; 7 2 0 0 0 0
%e n=6; 11 5 2 0 0 0 0
%e n=7; 15 8 4 0 0 0 0 0
%t (* 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}]];
%t a2=Map[{First[#],First[#]>Length[\[Mu]]||\[Mu][[First[#]]]<#[[2]]}&,a1];a3=Map[First,DeleteCases[SplitBy[a2,MatchQ[#,{_,False}]&],{{_,False}}],{2}];
%t Flatten[redu[Part[\[Lambda],#], Part[PadRight[\[Mu],Length[\[Lambda]],0],#]/. 0->Sequence[]]&/@Map[Union,a3],1]];
%t 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}]
%Y Cf. A259478, A259479, A259481, A161492, A227309, A006958.
%K nonn,tabl
%O 0,4
%A _Wouter Meeussen_, Jul 01 2015
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