login
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.
13
1, 2, 1, 4, 3, 1, 10, 11, 4, 1, 26, 56, 23, 5, 1, 76, 348, 214, 42, 6, 1, 232, 2578, 2698, 641, 69, 7, 1, 764, 22054, 44288, 14751, 1620, 106, 8, 1, 2620, 213798, 902962, 478711, 62781, 3616, 154, 9, 1, 9496, 2313638, 22262244, 20758650, 3710272, 222190, 7340, 215, 10, 1
OFFSET
1,2
COMMENTS
From Andrew Howroyd, Apr 09 2020: (Start)
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.
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)
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..351 (first 95 terms from R. H. Hardin; terms 96..153 from Alois P. Heinz)
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014
EXAMPLE
Table starts
1 2 4 10 26 76 232 764 2620
1 3 11 56 348 2578 22054 213798 2313638
1 4 23 214 2698 44288 902962 22262244 648446612
1 5 42 641 14751 478711 20758650 1158207312 80758709676
1 6 69 1620 62781 3710272 313568636 36218801244 5518184697792
1 7 106 3616 222190 22393101 3444274966 767013376954 ...
1 8 154 7340 681460 111200600 29445929253 ...
1 9 215 13825 1865715 472211360 ...
1 10 290 24510 4655535 ...
1 11 381 41336 ...
...
All solutions for 4 X 2:
..1..0....1..1....1..1
..1..0....1..1....1..0
..0..1....0..0....0..1
..0..1....0..0....0..0
PROG
(PARI)
T(k, n)={
local(M=Map(Mat([0, 1])));
my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
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)))));
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]);
}
{for(n=1, 7, for(k=1, 7, print1(T(n, k), ", ")); print)} \\ Andrew Howroyd, Apr 08 2020
CROSSREFS
Columns 1..8 are A000012, A000027(n+1), A019298(n+1), A053493, A053494, A188400, A188401, A188402.
Main diagonal is A333739.
Sequence in context: A067410 A213947 A348482 * A248929 A109977 A201199
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Mar 30 2011
STATUS
approved