A316587 -id:A316587 - cp's OEIS frontend

1, 3, 10, 20, 43, 75, 132, 208, 325, 475, 686, 948, 1295, 1715, 2248, 2880, 3657, 4563, 5650, 6900, 8371, 10043, 11980, 14160, 16653, 19435, 22582, 26068, 29975, 34275, 39056, 44288, 50065, 56355, 63258, 70740, 78907, 87723, 97300, 107600, 118741, 130683, 143550, 157300

Offset: 0

Yu Hin Au, Jul 24 2020

Let S be a fixed matching of size 2 in a complete n-uniform hypergraph G with >= 4n vertices. Given T,T' (each also a matching of size 2), define the equivalence relation where T ~ T' if and only if there exists an automorphism of G that maps every hyperedge in T to a hyperedge in T' while mapping every hyperedge in S to a hyperedge in S. Then the number of equivalence classes is a(n).

a(n) is the number of equivalence classes of 2 X 2 matrices with nonnegative integer entries where each row and column sum to at most n, such that two matrices are related if one can be obtained from the other by permuting rows and columns.

			To see a(2)=10, let S = {{1,2},{3,4}}. Then a representative from each of the 10 equivalence classes are
  1. {{1,2}, {3,4}}
  2. {{1,3}, {2,4}}
  3. {{1,5}, {3,4}}
  4. {{1,3}, {4,5}}
  5. {{1,2}, {5,6}}
  6. {{1,3}, {5,6}}
  7. {{1,5}, {2,6}}
  8. {{1,5}, {3,6}}
  9. {{1,5}, {6,7}}
  10. {{5,6}, {7,8}}
Likewise, in the 2 X 2 matrix interpretation, a representative from each of the a(2)=10 equivalence classes are
  [2 0 ; 0 2]
  [1 1 ; 1 1]
  [2 0 ; 0 1]
  [1 1 ; 1 0]
  [2 0 ; 0 0]
  [1 1 ; 0 0]
  [1 0 ; 1 0]
  [1 0 ; 0 1]
  [1 0 ; 0 0]
  [0 0 ; 0 0]

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Yu Hin Au, Nathan Lindzey, and Levent Tunçel, Matchings, hypergraphs, association schemes, and semidefinite optimization, arXiv:2008.08628 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A316587.

Nest[Append[#1, (#2^3 + 5 #2 + 3)/3 + 2*Floor[#2/2] + #1[[-2]] ] & @@ {#, Length@ #} &, {1, 3}, 42] (* Michael De Vlieger, Nov 04 2020 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{1,3,10,20,43,75,132},60] (* Harvey P. Dale, May 28 2021 *)

Vec((1 + 2*x^2 - 2*x^3 + 3*x^4) / ((1 - x)^5*(1 + x)^2) + O(x^40)) \\ Colin Barker, Nov 05 2020

a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7). - Wesley Ivan Hurt, Nov 04 2020

G.f.: (1 + 2*x^2 - 2*x^3 + 3*x^4) / ((1 - x)^5*(1 + x)^2). - Colin Barker, Nov 05 2020

cp's OEIS Frontend

A336529 a(n) = (n^3+5n+3)/3 + 2floor(n/2) + a(n-2), with a(0)=1 and a(1)=3.

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