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A167625 Square array T(n,k), read by upward antidiagonals, counting isomorphism classes of k-regular multigraphs of order n, loops allowed.

%I A167625 #17 Mar 23 2020 15:44:45
%S A167625 1,1,0,1,1,1,1,0,2,0,1,1,3,2,1,1,0,5,0,3,0,1,1,7,8,7,3,1,1,0,11,0,20,
%T A167625 0,4,0,1,1,15,31,56,32,13,4,1,1,0,22,0,187,0,66,0,5,0,1,1,30,140,654,
%U A167625 727,384,101,22,5,1,1,0,42,0,2705,0,3369,0,181,0,6,0,1,1,56,722,12587,42703
%N A167625 Square array T(n,k), read by upward antidiagonals, counting isomorphism classes of k-regular multigraphs of order n, loops allowed.
%C A167625 The number of vertices n is positive; valency k is nonnegative.
%C A167625 Each loop contributes two to the valency of its vertex.
%C A167625 The antidiagonal having coordinate sum t=n+k is read from T(t,0) to T(1,t-1).
%C A167625 Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A333467. Burnside's lemma can be used to extend this method to the unlabeled case. - _Andrew Howroyd_, Mar 23 2020
%H A167625 Andrew Howroyd, <a href="/A167625/b167625.txt">Table of n, a(n) for n = 1..378</a> (27 antidiagonals, first 19 antidiagonals from Jason Kimberley)
%H A167625 J. S. Kimberley, <a href="http://oeis.org/wiki/User:Jason_Kimberley/A167625">Table in user subpage of wiki</a>.
%H A167625 R. C. Read, <a href="http://dx.doi.org/10.1112/jlms/s1-34.4.417">The enumeration of locally restricted graphs (I)</a>, J. London Math. Soc. 34 (1959) 417-436.
%F A167625 T(n,k) = N\{S_n[S_k] * S_{nk/2}[S_2]\}.
%e A167625 Array begins:
%e A167625 ==============================================
%e A167625 n\k | 0 1  2   3    4     5      6       7
%e A167625 ----+-----------------------------------------
%e A167625   1 | 1 0  1   0    1     0      1       0 ...
%e A167625   2 | 1 1  2   2    3     3      4       4 ...
%e A167625   3 | 1 0  3   0    7     0     13       0 ...
%e A167625   4 | 1 1  5   8   20    32     66     101 ...
%e A167625   5 | 1 0  7   0   56     0    384       0 ...
%e A167625   6 | 1 1 11  31  187   727   3369   12782 ...
%e A167625   7 | 1 0 15   0  654     0  40365       0 ...
%e A167625   8 | 1 1 22 140 2705 42703 675368 8584767 ...
%e A167625   ...
%Y A167625 Column sequences: A000012 (k=0), A059841 (k=1), A000041 (k=2), A129427 (k=3), A129429 (k=4), A129431 (k=5), A129433 (k=6), A129435 (k=7), A129437 (k=8).
%Y A167625 Cf. A333330 (loopless), A333397 (connected), A333467 (labeled).
%K A167625 nonn,tabl
%O A167625 1,9
%A A167625 _Jason Kimberley_, Nov 07 2009