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A188403 T(n,k) = Number of (n*k) X k binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.

%I A188403 #22 Apr 09 2020 14:25:35
%S A188403 1,2,1,4,3,1,10,11,4,1,26,56,23,5,1,76,348,214,42,6,1,232,2578,2698,
%T A188403 641,69,7,1,764,22054,44288,14751,1620,106,8,1,2620,213798,902962,
%U A188403 478711,62781,3616,154,9,1,9496,2313638,22262244,20758650,3710272,222190,7340,215,10,1
%N A188403 T(n,k) = Number of (n*k) X k binary arrays with rows in nonincreasing order, n ones in every column and no more than 2 ones in any row.
%C A188403 From _Andrew Howroyd_, Apr 09 2020: (Start)
%C A188403 T(n,k) is the number of k X k symmetric matrices with nonnegative integer entries and all row and column sums n. The number of such matrices up to isomorphism is given in A333737.
%C A188403 T(n,k) is also the number of loopless multigraphs with k labeled nodes of degree n or less. The number of such multigraphs up to isomorphism is given in A333893. (End)
%H A188403 Andrew Howroyd, <a href="/A188403/b188403.txt">Table of n, a(n) for n = 1..351</a> (first 95 terms from R. H. Hardin; terms 96..153 from Alois P. Heinz)
%H A188403 J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962, 2014
%e A188403 Table starts
%e A188403   1  2   4    10      26        76         232          764          2620
%e A188403   1  3  11    56     348      2578       22054       213798       2313638
%e A188403   1  4  23   214    2698     44288      902962     22262244     648446612
%e A188403   1  5  42   641   14751    478711    20758650   1158207312   80758709676
%e A188403   1  6  69  1620   62781   3710272   313568636  36218801244 5518184697792
%e A188403   1  7 106  3616  222190  22393101  3444274966 767013376954 ...
%e A188403   1  8 154  7340  681460 111200600 29445929253 ...
%e A188403   1  9 215 13825 1865715 472211360 ...
%e A188403   1 10 290 24510 4655535 ...
%e A188403   1 11 381 41336 ...
%e A188403   ...
%e A188403 All solutions for 4 X 2:
%e A188403 ..1..0....1..1....1..1
%e A188403 ..1..0....1..1....1..0
%e A188403 ..0..1....0..0....0..1
%e A188403 ..0..1....0..0....0..0
%o A188403 (PARI)
%o A188403 T(k,n)={
%o A188403   local(M=Map(Mat([0, 1])));
%o A188403   my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
%o A188403   my(recurse(r, h, p, q, v, e) = if(!p, acc(x^e+q, v), my(i=poldegree(p), t=pollead(p)); self()(r, k, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, (k-e)\m), self()(r, if(j==t, k, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e+j*m)))));
%o A188403   for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-r, k, src[i, 1], 0, src[i, 2], 0))); vecsum(Mat(M)[,2]);
%o A188403 }
%o A188403 {for(n=1, 7, for(k=1, 7, print1(T(n,k),", ")); print)} \\ _Andrew Howroyd_, Apr 08 2020
%Y A188403 Columns 1..8 are A000012, A000027(n+1), A019298(n+1), A053493, A053494, A188400, A188401, A188402.
%Y A188403 Rows 1..8 are A000085, A000985, A188404, A188405, A188406, A188407, A188408, A188409.
%Y A188403 Main diagonal is A333739.
%Y A188403 Cf. A257493, A333157, A333737, A333893.
%K A188403 nonn,tabl
%O A188403 1,2
%A A188403 _R. H. Hardin_, Mar 30 2011