A358247 - OEIS

%I #21 Dec 01 2022 10:22:51

%S 1,8,28,71,132,217,309,417,521,638,746,866,975,1096,1205,1326,1435,

%T 1556,1665,1786,1895,2016,2125,2246,2355,2476,2585,2706,2815,2936,

%U 3045,3166,3275,3396,3505,3626,3735,3856,3965,4086,4195,4316,4425,4546,4655

%N Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 7, 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="/A358247/b358247.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) = 8 such pseudographs are: 1. two vertices connected by a 7-edge and a 0-edge, 2. two vertices connected by a 6-edge and a 1-edge, 3. two vertices connected by a 5-edge and a 2-edge, 4. two vertices connected by a 4-edge and a 3-edge, 5. two vertices where one has a 7-loop and the other one has a 0-loop, 6. two vertices where one has a 6-loop and the other one has a 1-loop, 7. two vertices where one has a 5-loop and the other one has a 2-loop, 8. two vertices where one has a 4-loop and the other one has a 3-loop.

%o (Julia)

%o using Combinatorics

%o function A(n::Int)

%o     sum_total = 7

%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 1:45])

%Y Other total edge weights: 3 (A358243), 4 (A358244), 5 (A358245), 6 (A358246), 8 (A358248), 9 (A358249).

%K nonn

%O 1,2

%A _Lars Göttgens_, Nov 04 2022