cp's OEIS Frontend

A263293 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n vertices and maximum vertex degree k, (0 <= k < n).

%I A263293 #63 Feb 16 2025 08:33:27
%S A263293 1,1,1,1,1,2,1,2,4,4,1,2,8,12,11,1,3,15,43,60,34,1,3,25,121,360,378,
%T A263293 156,1,4,41,378,2166,4869,3843,1044,1,4,65,1095,14306,68774,113622,
%U A263293 64455,12346,1,5,100,3441,104829,1141597,3953162,4605833,1921532,274668
%N A263293 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n vertices and maximum vertex degree k, (0 <= k < n).
%C A263293 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 A327366. Burnside's lemma can be used to extend this method to the unlabeled case. - _Andrew Howroyd_, Mar 10 2020
%H A263293 Andrew Howroyd, <a href="/A263293/b263293.txt">Table of n, a(n) for n = 1..210</a> (first 20 rows)
%H A263293 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000171">The degree of a graph</a>
%H A263293 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumVertexDegree.html">Maximum Vertex Degree</a>
%F A263293 From _Geoffrey Critzer_, Sep 10 2016: (Start)
%F A263293 G.f. for column k=0: A(x)=1/(1-x).
%F A263293 G.f. for column k=1: B(x)=x^2/((1-x^2)(1-x)).
%F A263293 G.f. for column k=2: 1/((1-x)(1-x^2))*Product_{i>=3} 1/(1-x^i)^2 - B(x) - A(x).
%F A263293 (End)
%F A263293 T(n, 0) = 1.
%F A263293 T(n, n - 1) = A000088(n - 1).
%F A263293 T(n, k) = A294217(n, n - 1 - k). - _Andrew Howroyd_, Sep 03 2019
%e A263293 Triangle begins:
%e A263293 1,
%e A263293 1,    1,
%e A263293 1,    1,    2,
%e A263293 1,    2,    4,    4,
%e A263293 1,    2,    8,   12,   11,
%e A263293 1,    3,   15,   43,   60,   34,
%e A263293 1,    3,   25,  121,  360,  378,  156,
%e A263293 1,    4,   41,  378, 2166, 4869, 3843, 1044,
%e A263293 ...
%Y A263293 Row sums are A000088 (simple graphs on n nodes).
%Y A263293 Column k=2 is A324740.
%Y A263293 Diagonals include A000088(n-1), A324693, A324670.
%Y A263293 Cf. A294217 (triangle of n-node minimum vertex degree counts).
%Y A263293 Cf. A327366.
%K A263293 nonn,tabl,nice
%O A263293 1,6
%A A263293 _Christian Stump_, Oct 13 2015
%E A263293 Rows n=9 and 10 added by _Eric W. Weisstein_, Oct 24 2017