cp's OEIS Frontend

A358245 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 5, up to isomorphism.

%I A358245 #26 Jan 01 2023 15:58:27
%S A358245 1,6,17,36,59,87,114,145,173,205,233,265,293,325,353,385,413,445,473,
%T A358245 505,533,565,593,625,653,685,713,745,773,805,833,865,893,925,953,985,
%U A358245 1013,1045,1073,1105,1133,1165,1193,1225,1253,1285,1313,1345,1373,1405,1433
%N A358245 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 5, up to isomorphism.
%C A358245 Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.
%H A358245 Lars Göttgens, <a href="/A358245/b358245.txt">Table of n, a(n) for n = 1..10000</a>
%H A358245 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 A358245 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pseudograph.html">Pseudograph</a>.
%F A358245 Apparently a(n) = a(n-1) + a(n-2) - a(n-3) for n >= 11. - _Hugo Pfoertner_, Dec 02 2022
%e A358245 For n = 2 the a(2) = 6 such pseudographs are: 1. two vertices connected by a 5-edge and a 0-edge, 2. two vertices connected by a 4-edge and a 1-edge, 3. two vertices connected by a 3-edge and a 2-edges, 4. two vertices where one has a 5-loop and the other one has a 0-loop, 5. two vertices where one has a 4-loop and the other one has a 1-loop, 6. two vertices where one has a 3-loop and the other one has a 2-loop.
%o A358245 (Julia)
%o A358245 using Combinatorics
%o A358245 function A(n::Int)
%o A358245     sum_total = 5
%o A358245     result = 0
%o A358245     for num_loops in 0:div(n, 2)
%o A358245         num_cross = n - 2 * num_loops
%o A358245         for sum_cross in 0:sum_total
%o A358245             for sum_loop1 in 0:sum_total-sum_cross
%o A358245                 sum_loop2 = sum_total - sum_cross - sum_loop1
%o A358245                 if sum_loop2 == sum_loop1
%o A358245                     result +=
%o A358245                         div(
%o A358245                             npartitions_with_zero(sum_loop2, num_loops) *
%o A358245                             (npartitions_with_zero(sum_loop2, num_loops) + 1),
%o A358245                             2,
%o A358245                         ) * npartitions_with_zero(sum_cross, num_cross)
%o A358245                 elseif sum_loop2 > sum_loop1
%o A358245                     result +=
%o A358245                         npartitions_with_zero(sum_loop2, num_loops) *
%o A358245                         npartitions_with_zero(sum_loop1, num_loops) *
%o A358245                         npartitions_with_zero(sum_cross, num_cross)
%o A358245                 end
%o A358245             end
%o A358245         end
%o A358245     end
%o A358245     return result
%o A358245 end
%o A358245 function npartitions_with_zero(n::Int, m::Int)
%o A358245     if m == 0
%o A358245         if n == 0
%o A358245             return 1
%o A358245         else
%o A358245             return 0
%o A358245         end
%o A358245     else
%o A358245         return Combinatorics.npartitions(n + m, m)
%o A358245     end
%o A358245 end
%o A358245 print([A(n) for n in 1:51])
%Y A358245 Other total edge weights: A358243 (3), A358244 (4), A358246 (6), A358247 (7), A358248 (8), A358249 (9).
%K A358245 nonn
%O A358245 1,2
%A A358245 _Lars Göttgens_, Nov 04 2022