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%I #18 Dec 01 2022 10:23:33
%S 1,10,35,99,190,332,484,680,863,1082,1277,1505,1704,1935,2135,2367,
%T 2567,2799,2999,3231,3431,3663,3863,4095,4295,4527,4727,4959,5159,
%U 5391,5591,5823,6023,6255,6455,6687,6887,7119,7319,7551,7751,7983,8183,8415,8615,8847
%N Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 8, up to isomorphism.
%C Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.
%H Lars Göttgens, <a href="/A358248/b358248.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pseudograph.html">Pseudograph</a>.
%e For n = 2 the a(2) = 10 such pseudographs are: 1. two vertices connected by a 8-edge and a 0-edge, 2. two vertices connected by a 7-edge and a 1-edge, 3. two vertices connected by a 6-edge and a 2-edge, 4. two vertices connected by a 5-edge and a 3-edge, 5. two vertices connected by two 4-edges, 6. two vertices where one has a 8-loop and the other one has a 0-loop, 7. two vertices where one has a 7-loop and the other one has a 1-loop, 8. two vertices where one has a 6-loop and the other one has a 2-loop, 9. two vertices where one has a 5-loop and the other one has a 3-loop, 10. two vertices with a 4-loop each.
%o (Julia)
%o using Combinatorics
%o function A(n::Int)
%o sum_total = 8
%o result = 0
%o for num_loops in 0:div(n, 2)
%o num_cross = n - 2 * num_loops
%o for sum_cross in 0:sum_total
%o for sum_loop1 in 0:sum_total-sum_cross
%o sum_loop2 = sum_total - sum_cross - sum_loop1
%o if sum_loop2 == sum_loop1
%o result +=
%o div(
%o npartitions_with_zero(sum_loop2, num_loops) *
%o (npartitions_with_zero(sum_loop2, num_loops) + 1),
%o 2,
%o ) * npartitions_with_zero(sum_cross, num_cross)
%o elseif sum_loop2 > sum_loop1
%o result +=
%o npartitions_with_zero(sum_loop2, num_loops) *
%o npartitions_with_zero(sum_loop1, num_loops) *
%o npartitions_with_zero(sum_cross, num_cross)
%o end
%o end
%o end
%o end
%o return result
%o end
%o function npartitions_with_zero(n::Int, m::Int)
%o if m == 0
%o if n == 0
%o return 1
%o else
%o return 0
%o end
%o else
%o return Combinatorics.npartitions(n + m, m)
%o end
%o end
%o print([A(n) for n in 1:46])
%Y Other total edge weights: 3 (A358243), 4 (A358244), 5 (A358245), 6 (A358246), 7 (A358247), 9 (A358249).
%K nonn
%O 1,2
%A _Lars Göttgens_, Nov 04 2022
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