cp's OEIS Frontend

A358249 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 9, up to isomorphism.

%I A358249 #18 Dec 01 2022 10:23:28
%S A358249 1,10,42,123,259,469,721,1034,1359,1726,2082,2472,2840,3239,3611,4013,
%T A358249 4386,4789,5162,5565,5938,6341,6714,7117,7490,7893,8266,8669,9042,
%U A358249 9445,9818,10221,10594,10997,11370,11773,12146,12549,12922,13325,13698,14101,14474
%N A358249 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 9, up to isomorphism.
%C A358249 Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.
%H A358249 Lars Göttgens, <a href="/A358249/b358249.txt">Table of n, a(n) for n = 1..10000</a>
%H A358249 J. Flake and V. Mackscheidt, <a href="https://arxiv.org/abs/2206.08226">Interpolating PBW Deformations for the Orthosymplectic Groups</a>, arXiv:2206.08226 [math.RT], 2022.
%H A358249 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pseudograph.html">Pseudograph</a>.
%e A358249 For n = 2 the a(2) = 10 such pseudographs are: 1. two vertices connected by a 9-edge and a 0-edge, 2. two vertices connected by a 8-edge and a 1-edge, 3. two vertices connected by a 7-edge and a 2-edge, 4. two vertices connected by a 6-edge and a 3-edge, 5. two vertices connected by a 5-edge and a 4-edge, 6. two vertices where one has a 9-loop and the other one has a 0-loop, 7. two vertices where one has a 8-loop and the other one has a 1-loop, 8. two vertices where one has a 7-loop and the other one has a 2-loop, 9. two vertices where one has a 6-loop and the other one has a 3-loop, 10. two vertices where one has a 5-loop and the other one has a 4-loop.
%o A358249 (Julia)
%o A358249 using Combinatorics
%o A358249 function A(n::Int)
%o A358249     sum_total = 9
%o A358249     result = 0
%o A358249     for num_loops in 0:div(n, 2)
%o A358249         num_cross = n - 2 * num_loops
%o A358249         for sum_cross in 0:sum_total
%o A358249             for sum_loop1 in 0:sum_total-sum_cross
%o A358249                 sum_loop2 = sum_total - sum_cross - sum_loop1
%o A358249                 if sum_loop2 == sum_loop1
%o A358249                     result +=
%o A358249                         div(
%o A358249                             npartitions_with_zero(sum_loop2, num_loops) *
%o A358249                             (npartitions_with_zero(sum_loop2, num_loops) + 1),
%o A358249                             2,
%o A358249                         ) * npartitions_with_zero(sum_cross, num_cross)
%o A358249                 elseif sum_loop2 > sum_loop1
%o A358249                     result +=
%o A358249                         npartitions_with_zero(sum_loop2, num_loops) *
%o A358249                         npartitions_with_zero(sum_loop1, num_loops) *
%o A358249                         npartitions_with_zero(sum_cross, num_cross)
%o A358249                 end
%o A358249             end
%o A358249         end
%o A358249     end
%o A358249     return result
%o A358249 end
%o A358249 function npartitions_with_zero(n::Int, m::Int)
%o A358249     if m == 0
%o A358249         if n == 0
%o A358249             return 1
%o A358249         else
%o A358249             return 0
%o A358249         end
%o A358249     else
%o A358249         return Combinatorics.npartitions(n + m, m)
%o A358249     end
%o A358249 end
%o A358249 print([A(n) for n in 1:43])
%Y A358249 Other total edge weights: 3 (A358243), 4 (A358244), 5 (A358245), 6 (A358246), 7 (A358247), 8 (A358248).
%K A358249 nonn
%O A358249 1,2
%A A358249 _Lars Göttgens_, Nov 04 2022