A358245 Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 5, up to isomorphism.

1, 6, 17, 36, 59, 87, 114, 145, 173, 205, 233, 265, 293, 325, 353, 385, 413, 445, 473, 505, 533, 565, 593, 625, 653, 685, 713, 745, 773, 805, 833, 865, 893, 925, 953, 985, 1013, 1045, 1073, 1105, 1133, 1165, 1193, 1225, 1253, 1285, 1313, 1345, 1373, 1405, 1433

Offset: 1

Author

Lars Göttgens, Nov 04 2022

Keywords

• nonn

Comments

Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.

Examples

			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.
		

Links

• Lars Göttgens, Table of n, a(n) for n = 1..10000
• J. Flake and V. Mackscheidt, Interpolating PBW Deformations for the Orthosymplectic Groups, arXiv:2206.08226 [math.RT], 2022.
• Eric Weisstein's World of Mathematics, Pseudograph.

Crossrefs

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

Programs

Julia

using Combinatorics
function A(n::Int)
    sum_total = 5
    result = 0
    for num_loops in 0:div(n, 2)
        num_cross = n - 2 * num_loops
        for sum_cross in 0:sum_total
            for sum_loop1 in 0:sum_total-sum_cross
                sum_loop2 = sum_total - sum_cross - sum_loop1
                if sum_loop2 == sum_loop1
                    result +=
                        div(
                            npartitions_with_zero(sum_loop2, num_loops) *
                            (npartitions_with_zero(sum_loop2, num_loops) + 1),
                            2,
                        ) * npartitions_with_zero(sum_cross, num_cross)
                elseif sum_loop2 > sum_loop1
                    result +=
                        npartitions_with_zero(sum_loop2, num_loops) *
                        npartitions_with_zero(sum_loop1, num_loops) *
                        npartitions_with_zero(sum_cross, num_cross)
                end
            end
        end
    end
    return result
end
function npartitions_with_zero(n::Int, m::Int)
    if m == 0
        if n == 0
            return 1
        else
            return 0
        end
    else
        return Combinatorics.npartitions(n + m, m)
    end
end
print([A(n) for n in 1:51])

Formula

Apparently a(n) = a(n-1) + a(n-2) - a(n-3) for n >= 11. - Hugo Pfoertner, Dec 02 2022