%I #22 Apr 09 2020 14:25:35

%S 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 641,69,7,1,764,22054,44288,14751,1620,106,8,1,2620,213798,902962,

%U 478711,62781,3616,154,9,1,9496,2313638,22262244,20758650,3710272,222190,7340,215,10,1

%N 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 From _Andrew Howroyd_, Apr 09 2020: (Start)

%C 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 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 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 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 Table starts

%e 1 2 4 10 26 76 232 764 2620

%e 1 3 11 56 348 2578 22054 213798 2313638

%e 1 4 23 214 2698 44288 902962 22262244 648446612

%e 1 5 42 641 14751 478711 20758650 1158207312 80758709676

%e 1 6 69 1620 62781 3710272 313568636 36218801244 5518184697792

%e 1 7 106 3616 222190 22393101 3444274966 767013376954 ...

%e 1 8 154 7340 681460 111200600 29445929253 ...

%e 1 9 215 13825 1865715 472211360 ...

%e 1 10 290 24510 4655535 ...

%e 1 11 381 41336 ...

%e ...

%e All solutions for 4 X 2:

%e ..1..0....1..1....1..1

%e ..1..0....1..1....1..0

%e ..0..1....0..0....0..1

%e ..0..1....0..0....0..0

%o (PARI)

%o T(k,n)={

%o local(M=Map(Mat([0, 1])));

%o my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));

%o 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 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 }

%o {for(n=1, 7, for(k=1, 7, print1(T(n,k),", ")); print)} \\ _Andrew Howroyd_, Apr 08 2020

%Y Columns 1..8 are A000012, A000027(n+1), A019298(n+1), A053493, A053494, A188400, A188401, A188402.

%Y Rows 1..8 are A000085, A000985, A188404, A188405, A188406, A188407, A188408, A188409.

%Y Main diagonal is A333739.

%Y Cf. A257493, A333157, A333737, A333893.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Mar 30 2011