A002727 -id:A002727 - cp's OEIS frontend

A370081 Number of left-labeled (3,n)-bipartite graphs.

1, 8, 25, 60, 129, 256, 484, 872, 1512, 2532, 4110, 6484, 9975, 14992, 22065, 31860, 45207, 63126, 86868, 117934, 158130, 209600, 274874, 356916, 459189, 585694, 741053, 930566, 1160285, 1437090, 1768784, 2164158, 2633108, 3186722, 3837386, 4598894, 5486579, 6517410, 7710149

Offset: 0

Yavuz Oruc, Feb 08 2024

nonn

a(n) = number of left-labeled (3,n)-bipartite graphs. The bipartite graphs counted here arise as representations of certain types of calls in switching networks in which three callers can be in a call with an arbitrary number (n) of receivers, and where callers are distinguishable, but the receivers are not. In a more abstract setting, a left-labeled (3,n)-bipartite graph is a graph with two sets of non-overlapping vertices I and O, where |I| = 3, |O| = n, and the vertices in I are considered different (distinguishable) up to subsets, whereas the n vertices in O are considered entirely interchangeable (indistinguishable). The sequence gives the number of non-isomorphic graphs under these assumptions.

			For n = 1, a(1) = 8 and there are 8 left-labeled (3,1)-bipartite graphs. Suppose the left vertices are labeled a, b, c and the right vertex is labeled d. The left-labeled (3,1)-bipartite graphs are:
(1) Empty bipartite graph (no edges)
(2) Place an edge between a and d.
(3) Place an edge between b and d.
(4) Place an edge between c and d.
(5) Place an edge between a and d, and b and d.
(6) Place an edge between a and d, and c and d.
(7) Place an edge between b and d, and c and d.
(8) Place an edge between a and d, b and d, and c and d.
For n = 2, a(2) = 25 and there are 25 left-labeled (3,2)-bipartite graphs. These left-labeled (3,2)-bipartite graphs are listed in the publication that is given in the reference section.

Yavuz Oruc, Table of n, a(n) for n = 0..50
A. Atmaca and A. Yavuz Oruc, On The Number Of Labeled Bipartite Graphs, arXiv:2402.08053 [math.CO], 2024.

Cf. A002727.

ct1[n_] :=
  Binomial[n+7,7]+(3*(n+4)*(2*n^4+32*n^3+172*n^2+352*n+15*(-1)^n+225))/960;
ct2[n_] := (4*n^3+30*n^2+68*n-3+3*(-1)^n)/24;
B3rmod0 = Function[n,(1/6)(ct1[n] + (n^3 + 12*n^2 + 45*n + 54)/27)+ct2[n]] /@Range[0,50,3];
B3rmod1 = Function[n,(1/6)(ct1[n] + (n^3 + 12*n^2 + 45*n + 50)/27)+ct2[n]] /@Range[1,50,3];
B3rmod2 = Function[n,(1/6)(ct1[n] + (n^3 + 12*n^2 + 39*n + 28)/27)+ct2[n]] /@Range[2,50,3];
B3r = {B3rmod0, B3rmod1, B3rmod2}~Flatten~{2, 1}

Let
  ct1(n) = binomial(n+7,7) + ((n+4)*(2*n^4 + 32*n^3 + 172*n^2 + 352*n + 15*(-1)^n + 225))/320,
  ct2(n) = (4*n^3 + 30*n^2 + 68*n - 3 + 3*(-1)^n)/24,
and
  f(n) = 0            if n mod 3 = 0
       = 2/81         if n mod 3 = 1
       = n/27 + 13/81 if n mod 3 = 2
Then
  a(n) = (1/6)*(ct1(n) + (n^3+12*n^2+45*n+54)/27)+ct2(n)-f(n)

A052266 Number of 3 X n matrices over GF(3) under row and column permutations.

Original entry on oeis.org

1, 10, 92, 738, 5053, 29930, 155904, 725560, 3061223, 11854073, 42564874, 142944110, 452197071, 1355697097, 3871729568, 10579613832, 27765888553, 70221521921, 171635652958, 406476831574, 934849110107, 2092203571359

Offset: 0

Vladeta Jovovic, Feb 04 2000

nonn

Cf. A002727.

G.f.: (x^24 + 28*x^23 + 148*x^22 + 518*x^21 + 1862*x^20 + 5096*x^19 + 11172*x^18 + 21813*x^17 + 37333*x^16 + 53936*x^15 + 68880*x^14 + 79114*x^13 + 79184*x^12 + 68880*x^11 + 53992*x^10 + 37277*x^9 + 21813*x^8 + 11144*x^7 + 5124*x^6 + 1862*x^5 + 526*x^4 + 140*x^3 + 28*x^2 + 1)/((1 - x^3)^8*(1 - x^2)^9*(1 - x)^10).

A055536 Number of asymmetric types of (3,n)-hypergraphs under action of symmetric group S_3.

Original entry on oeis.org

1, 8, 29, 82, 198, 426, 841, 1556, 2726, 4568, 7373, 11522, 17507, 25958, 37658, 53582, 74924, 103130, 139938, 187426, 248044, 324678, 420698, 540014, 687141, 867274, 1086343, 1351108, 1669234, 2049376, 2501275, 3035866, 3665362

Offset: 2

Vladeta Jovovic, Jul 09 2000

nonn

			There are 8 asymmetric (3,3)-hypergraphs: {{1,2},{1,2},{1,3}}, {{1},{1,2},{1,2,3}}, {{1},{1,2},{1,2}}, {{1},{1},{1,2}}, {{1},{1,2},{2,3}}, {{1},{2},{1,3}}, {{1},{1},{2}}, {0,{1},{1,2}}.

Cf. A002727.

G.f.: (1/(1-x)^8-3/(1-x)^4/(1-x^2)^2+2/(1-x)^2/(1-x^3)^2)/6.

More terms from James Sellers, Jul 11 2000

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A370081 Number of left-labeled (3,n)-bipartite graphs.

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A052266 Number of 3 X n matrices over GF(3) under row and column permutations.

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A055536 Number of asymmetric types of (3,n)-hypergraphs under action of symmetric group S_3.

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