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A028657
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Triangle read by rows: T(n,k) = number of n-node graphs with k nodes in distinguished bipartite block, k = 0..n.
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34
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 22, 36, 22, 6, 1, 1, 7, 34, 87, 87, 34, 7, 1, 1, 8, 50, 190, 317, 190, 50, 8, 1, 1, 9, 70, 386, 1053, 1053, 386, 70, 9, 1, 1, 10, 95, 734, 3250, 5624, 3250, 734, 95, 10, 1, 1, 11, 125, 1324, 9343, 28576, 28576, 9343, 1324, 125, 11, 1
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OFFSET
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0,5
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COMMENTS
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Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color; the color classes are not interchangeable.
Also the number of principal transversal matroids (also known as fundamental transversal matroids) of size n and rank k (originally enumerated by Brylawski). - Gordon F. Royle, Oct 30 2007
This sequence is also obtained if we read the array A(m,n) = number of inequivalent m X n binary matrices by antidiagonals, where equivalence means permutations of rows or columns (m>=0, n>=0) [Kerber]. - N. J. A. Sloane, Sep 01 2013
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REFERENCES
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R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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LINKS
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A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
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FORMULA
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A(m,n) = Sum_{p in P(m), q in P(n)} 2^Sum_{i in p, j in q} gcd(i,j) / (N(p) N(q)) where P(m) are the partition of m (see e.g., A036036), N(p) = Sum_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p.
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EXAMPLE
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The triangle T(n,k) begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 7, 4, 1;
1, 5, 13, 13, 5, 1;
1, 6, 22, 36, 22, 6, 1;
...
For example, there are 36 graphs on 6 nodes with a distinguished bipartite block with 3 nodes.
The array A(m,n) (m>=0, n>=0) (see Comments) begins:
1 1 1 1 1 1 1 1 1 ...
1 2 3 4 5 6 7 8 9 ...
1 3 7 13 22 34 50 70 95 ...
1 4 13 36 87 190 386 734 1324 ...
1 5 22 87 317 1053 3250 9343 25207 ...
1 6 34 190 1053 5624 28576 136758 613894 ...
1 7 50 386 3250 28576 251610 2141733 17256831 ...
1 8 70 734 9343 136758 2141733 33642660 508147108 ...
1 9 95 1324 25207 613894 17256831 508147108 14685630688 ...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
{seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
end:
g:= proc(n, k) option remember; add(add(2^add(add(igcd(i, j)*
coeff(s, x, i)* coeff(t, x, j), j=1..degree(t)),
i=1..degree(s))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
i=1..degree(s))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
i=1..degree(t)), t=b(n+k$2)), s=b(n$2))
end:
A:= (n, k)-> g(min(n, k), abs(n-k)):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Union[ Flatten[ Table[ Function[ {p}, p + j*x^i] /@ b[n - i*j, i-1], {j, 0, n/i}]]]]];
g[n_, k_] := g[n, k] = Sum[ Sum[ 2^Sum[ Sum[GCD[i, j] * Coefficient[s, x, i] * Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[i^Coefficient[s, x, i] * Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i] * Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+k, n+k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n-k]];
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PROG
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(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={sum(j=1, #q, gcd(t, q[j]))}
A(n, m)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(exp(sum(t=1, n, 2^K(q, t)/t*x^t) + O(x*x^n)), n)); s/m!}
{ for(r=0, 10, for(k=0, r, print1(A(r-k, k), ", ")); print) } \\ Andrew Howroyd, Mar 25 2020
(PARI) \\ G(k, x) gives k-th column as rational function (see Jovovic link).
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
Fix(q, x)={my(v=divisors(lcm(Vec(q))), u=apply(t->2^sum(j=1, #q, gcd(t, q[j])), v)); 1/prod(i=1, #v, my(t=v[i]); (1-x^t)^(sum(j=1, i, my(d=t/v[j]); if(!frac(d), moebius(d)*u[j]))/t))}
G(m, x)={my(s=0); forpart(q=m, s+=permcount(q)*Fix(q, x)); s/m!}
T(n, k)={my(m=max(k, n-k)); polcoef(G(n-m, x + O(x*x^m)), m)} \\ Andrew Howroyd, Mar 26 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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