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Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 5, up to isomorphism.
7

%I #26 Jan 01 2023 15:58:27

%S 1,6,17,36,59,87,114,145,173,205,233,265,293,325,353,385,413,445,473,

%T 505,533,565,593,625,653,685,713,745,773,805,833,865,893,925,953,985,

%U 1013,1045,1073,1105,1133,1165,1193,1225,1253,1285,1313,1345,1373,1405,1433

%N Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 5, 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="/A358245/b358245.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>.

%F Apparently a(n) = a(n-1) + a(n-2) - a(n-3) for n >= 11. - _Hugo Pfoertner_, Dec 02 2022

%e 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 (Julia)

%o using Combinatorics

%o function A(n::Int)

%o sum_total = 5

%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 1:51])

%Y Other total edge weights: A358243 (3), A358244 (4), A358246 (6), A358247 (7), A358248 (8), A358249 (9).

%K nonn

%O 1,2

%A _Lars Göttgens_, Nov 04 2022