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A259480 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).

%I A259480 #27 Mar 02 2025 10:10:12
%S A259480 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 A259480 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 A259480 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 A259480 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 A259480 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 A259480 The diagonals T(n,n-k) count connected skew diagrams with weight k:
%C A259480   1; 2; 3,1; 5,2,2; 7,5,4,3,1; 11,8,10,7,6,2,2;
%C A259480 Their sums equal A006958.
%D A259480 I. G. MacDonald: "Symmetric functions and Hall polynomials"; Oxford University Press, 1979. Page 4.
%H A259480 Wouter Meeussen, <a href="/A259480/a259480.txt">Table n,m, T(n,m) for n= 1..27</a>
%e A259480 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 A259480 The diagrams are:
%e A259480   x x 0 0 , x x 0 , x 0 0 , x 0
%e A259480   0 0 0     0 0 0   x 0     x 0
%e A259480             0       0 0     0 0
%e A259480                             0
%e A259480 Triangle begins:
%e A259480       k=0  1  2  3  4  5  6  7
%e A259480   n=0;  0
%e A259480   n=1;  1  0
%e A259480   n=2;  2  0  0
%e A259480   n=3;  3  0  0  0
%e A259480   n=4;  5  1  0  0  0
%e A259480   n=5;  7  2  0  0  0  0
%e A259480   n=6; 11  5  2  0  0  0  0
%e A259480   n=7; 15  8  4  0  0  0  0  0
%t A259480 (* 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 A259480 a2=Map[{First[#],First[#]>Length[\[Mu]]||\[Mu][[First[#]]]<#[[2]]}&,a1];a3=Map[First,DeleteCases[SplitBy[a2,MatchQ[#,{_,False}]&],{{_,False}}],{2}];
%t A259480 Flatten[redu[Part[\[Lambda],#], Part[PadRight[\[Mu],Length[\[Lambda]],0],#]/. 0->Sequence[]]&/@Map[Union,a3],1]];
%t A259480 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 A259480 Cf. A259478, A259479, A259481, A161492, A227309, A006958.
%K A259480 nonn,tabl
%O A259480 0,4
%A A259480 _Wouter Meeussen_, Jul 01 2015