cp's OEIS Frontend

A358244 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 4, up to isomorphism.

%I A358244 #42 Jan 03 2023 05:52:46
%S A358244 1,6,13,27,38,55,67,85,97,115,127,145,157,175,187,205,217,235,247,265,
%T A358244 277,295,307,325,337,355,367,385,397,415,427,445,457,475,487,505,517,
%U A358244 535,547,565,577,595,607,625,637,655,667,685,697,715,727,745,757,775
%N A358244 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 4, up to isomorphism.
%C A358244 Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.
%H A358244 Lars Göttgens, <a href="/A358244/b358244.txt">Table of n, a(n) for n = 1..10000</a>
%H A358244 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 A358244 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pseudograph.html">Pseudograph</a>.
%F A358244 Apparently a(n) = 6*A047209(n-2) + 1 for n >= 6, i.e., terms satisfy the linear recurrence a(n) = a(n-1) + a(n-2) - a(n-3) for n >= 9. - _Hugo Pfoertner_, Dec 02 2022
%e A358244 For n = 2 the a(2) = 6 such pseudographs are:
%e A358244   1. two vertices connected by a 4-edge and a 0-edge,
%e A358244   2. two vertices connected by a 3-edge and a 1-edge,
%e A358244   3. two vertices connected by two 2-edges,
%e A358244   4. two vertices where one has a 4-loop and the other one has a 0-loop,
%e A358244   5. two vertices where one has a 3-loop and the other one has a 1-loop,
%e A358244   6. two vertices with a 2-loop each.
%o A358244 (Julia)
%o A358244 using Combinatorics
%o A358244 function A(n::Int)
%o A358244     sum_total = 4
%o A358244     result = 0
%o A358244     for num_loops in 0:div(n, 2)
%o A358244         num_cross = n - 2 * num_loops
%o A358244         for sum_cross in 0:sum_total
%o A358244             for sum_loop1 in 0:sum_total-sum_cross
%o A358244                 sum_loop2 = sum_total - sum_cross - sum_loop1
%o A358244                 if sum_loop2 == sum_loop1
%o A358244                     result +=
%o A358244                         div(
%o A358244                             npartitions_with_zero(sum_loop2, num_loops) *
%o A358244                             (npartitions_with_zero(sum_loop2, num_loops) + 1),
%o A358244                             2,
%o A358244                         ) * npartitions_with_zero(sum_cross, num_cross)
%o A358244                 elseif sum_loop2 > sum_loop1
%o A358244                     result +=
%o A358244                         npartitions_with_zero(sum_loop2, num_loops) *
%o A358244                         npartitions_with_zero(sum_loop1, num_loops) *
%o A358244                         npartitions_with_zero(sum_cross, num_cross)
%o A358244                 end
%o A358244             end
%o A358244         end
%o A358244     end
%o A358244     return result
%o A358244 end
%o A358244 function npartitions_with_zero(n::Int, m::Int)
%o A358244     if m == 0
%o A358244         if n == 0
%o A358244             return 1
%o A358244         else
%o A358244             return 0
%o A358244         end
%o A358244     else
%o A358244         return Combinatorics.npartitions(n + m, m)
%o A358244     end
%o A358244 end
%o A358244 print([A(n) for n in 1:54])
%Y A358244 Other total edge weights: A358243 (3), A358245 (5), A358246 (6), A358247 (7), A358248 (8), A358249 (9).
%Y A358244 Cf. A047209.
%K A358244 nonn
%O A358244 1,2
%A A358244 _Lars Göttgens_, Nov 04 2022