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A259479 Skew diagrams, both connected or not.

%I A259479 #25 Mar 02 2025 10:10:16
%S A259479 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,
%T A259479 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,
%U A259479 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
%N A259479 Skew diagrams, both connected or not.
%C A259479 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.
%D A259479 I. G. MacDonald: "Symmetric functions and Hall polynomials", Oxford University Press, 1979. Page 4.
%H A259479 Wouter Meeussen, <a href="/A259479/a259479.txt">Table n, m, T(n,m) for n= 1..27</a>
%e A259479 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))
%e A259479 and the diagrams are:
%e A259479   x x 0 0 , x x 0 , x x 0 , x 0 0 , x 0 , x 0
%e A259479   0 0       0 0 0   0 0     x 0     x 0   x 0
%e A259479                     0       0       0 0   0
%e A259479                                           0
%e A259479 Triangle begins:
%e A259479       k=0  1  2  3  4  5  6
%e A259479   n=0;  1
%e A259479   n=1;  1  0
%e A259479   n=2;  2  0  0
%e A259479   n=3;  3  1  0  0
%e A259479   n=4;  5  3  0  0  0
%e A259479   n=5;  7  5  2  0  0  0
%e A259479   n=6; 11  9  6  1  0  0  0
%t A259479 majorsweak[left_List, right_List]:=Block[{le1=Length[left], le2=Length[right]}, If[le2>le1||Min[Sign[left-PadRight[right, le1]]]<0, False, True]];
%t A259479 redu1[\[Lambda]_,\[Mu]_]/;majorsweak[\[Lambda],\[Mu]]:=Delete[#,List/@DeleteCases[Table[i Boole[\[Lambda][[i]]==\[Mu][[i]]],{i,Length[\[Mu]]}],0]]&/@{\[Lambda],\[Mu]};
%t A259479 redu[\[Lambda]_,\[Mu]_]/;majorsweak[\[Lambda],\[Mu]]:=TransposePartition/@Apply[redu1,TransposePartition/@redu1[\[Lambda],\[Mu]]];
%t A259479 Table[Sum[Boole[majorsweak[\[Lambda],\[Mu]]&&redu[\[Lambda],\[Mu]]=={\[Lambda],\[Mu]}],{\[Lambda],Partitions[n]},{\[Mu],Partitions[k]}],{n,0,12},{k,0,n}];
%Y A259479 Cf. A259478, A259480, A259481, A161492, A227309, A006958.
%K A259479 nonn,tabl
%O A259479 0,4
%A A259479 _Wouter Meeussen_, Jun 28 2015