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A369283 Triangle read by rows: T(n,k) is the number of labeled point-determining graphs with n nodes and k edges, n >= 0, 0 <= k <= n*(n - 1)/2.

%I A369283 #11 Jan 20 2024 13:52:32
%S A369283 1,1,0,1,0,3,0,1,0,0,3,16,12,0,1,0,0,15,60,130,132,140,80,30,0,1,0,0,
%T A369283 0,15,600,1692,3160,4635,4620,3480,2088,885,240,60,0,1,0,0,0,105,1260,
%U A369283 7665,28042,74280,142380,218960,271404,276150,230860,157710,86250,38752,13524,3360,560,105,0,1
%N A369283 Triangle read by rows: T(n,k) is the number of labeled point-determining graphs with n nodes and k edges, n >= 0, 0 <= k <= n*(n - 1)/2.
%C A369283 Point-determining graphs are also called mating graphs.
%H A369283 Andrew Howroyd, <a href="/A369283/b369283.txt">Table of n, a(n) for n = 0..1350</a> (rows 0..20)
%H A369283 Ira M. Gessel and Ji Li, <a href="https://doi.org/10.1016/j.jcta.2010.03.009">Enumeration of point-determining graphs</a>, J. Combinatorial Theory Ser. A 118 (2011), 591-612.
%F A369283 Sum_{k>=0} 2^k*T(n,k) = A102596(n).
%F A369283 Sum_{k>=0} 3^k*T(n,k) = A102579(n).
%e A369283 Triangle begins:
%e A369283  [0] 1;
%e A369283  [1] 1;
%e A369283  [2] 0, 1;
%e A369283  [3] 0, 3,  0,  1;
%e A369283  [4] 0, 0,  3, 16,  12,    0,    1;
%e A369283  [5] 0, 0, 15, 60, 130,  132,  140,   80,   30,    0,    1;
%e A369283  [6] 0, 0,  0, 15, 600, 1692, 3160, 4635, 4620, 3480, 2088, 885, 240, 60, 0, 1;
%e A369283   ...
%o A369283 (PARI)
%o A369283 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}
%o A369283 edges(p,t) = {prod(i=2, #p, prod(j=1, i-1, t(p[i]*p[j])))}
%o A369283 row(n) = {my(s=0); forpart(p=n, s += permcount(p)*(-1)^(n-#p)*edges(p, w->1 + x^w)); Vecrev(s)}
%Y A369283 Row sums are A006024.
%Y A369283 Cf. A102579, A102596, A368987 (unlabeled).
%K A369283 nonn,tabf
%O A369283 0,6
%A A369283 _Andrew Howroyd_, Jan 18 2024