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Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 9, up to isomorphism.
7

%I #18 Dec 01 2022 10:23:28

%S 1,10,42,123,259,469,721,1034,1359,1726,2082,2472,2840,3239,3611,4013,

%T 4386,4789,5162,5565,5938,6341,6714,7117,7490,7893,8266,8669,9042,

%U 9445,9818,10221,10594,10997,11370,11773,12146,12549,12922,13325,13698,14101,14474

%N Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 9, 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="/A358249/b358249.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 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 (Julia)

%o using Combinatorics

%o function A(n::Int)

%o sum_total = 9

%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 1:43])

%Y Other total edge weights: 3 (A358243), 4 (A358244), 5 (A358245), 6 (A358246), 7 (A358247), 8 (A358248).

%K nonn

%O 1,2

%A _Lars Göttgens_, Nov 04 2022