|
%I #21 Oct 13 2023 11:28:55
%S 1,4,15,82,457,3231,24055,209375,1955288,20455936,229830841,
%T 2828166755,37228913365,528635368980,7990596990430,128909374528433,
%U 2202090635802581,39837079499488151,759320365206705013,15234890522990662422,320634889654149218205,7068984425261215971205
%N Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.
%C This is the sum over the matrix of base change from the elementary symmetric functions to the monomial symmetric functions.
%D I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford 1979, p. 57.
%H Ludovic Schwob, <a href="/A068313/b068313.txt">Table of n, a(n) for n = 1..37</a>
%e a(2) = 4 because there are 4 different 0-1 matrices of weight 2: 1 10 01 11,1, 01, 10.
%e From _Gus Wiseman_, Nov 15 2018: (Start)
%e The a(3) = 15 matrices:
%e [1 1 1]
%e .
%e [1 1] [1 1 0] [1 0 1] [0 1 1]
%e [1 0] [0 0 1] [0 1 0] [1 0 0]
%e .
%e [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
%e [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
%e [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
%e (End)
%t prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@T[prs2mat[#]]]]&]],{n,5}] (* _Gus Wiseman_, Nov 15 2018 *)
%Y Cf. A000219, A001970, A007716, A049311, A101370, A117433, A120733, A321646, A321652, A321653, A321654.
%K nonn
%O 1,2
%A Axel Kohnert (axel.kohnert(AT)uni-bayreuth.de), Feb 25 2002
%E Name changed by _Gus Wiseman_, Nov 15 2018
%E a(20) onwards from _Ludovic Schwob_, Oct 13 2023
|
