A336529 - OEIS

%I #23 May 28 2021 17:03:08

%S 1,3,10,20,43,75,132,208,325,475,686,948,1295,1715,2248,2880,3657,

%T 4563,5650,6900,8371,10043,11980,14160,16653,19435,22582,26068,29975,

%U 34275,39056,44288,50065,56355,63258,70740,78907,87723,97300,107600,118741,130683,143550,157300

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

%C 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).

%C 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.

%H Michael De Vlieger, <a href="/A336529/b336529.txt">Table of n, a(n) for n = 0..10000</a>

%H Yu Hin Au, Nathan Lindzey, and Levent Tunçel, <a href="https://arxiv.org/abs/2008.08628">Matchings, hypergraphs, association schemes, and semidefinite optimization</a>, arXiv:2008.08628 [math.CO], 2020.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).

%F 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

%F G.f.: (1 + 2*x^2 - 2*x^3 + 3*x^4) / ((1 - x)^5*(1 + x)^2). - _Colin Barker_, Nov 05 2020

%e To see a(2)=10, let S = {{1,2},{3,4}}. Then a representative from each of the 10 equivalence classes are

%e   1. {{1,2}, {3,4}}

%e   2. {{1,3}, {2,4}}

%e   3. {{1,5}, {3,4}}

%e   4. {{1,3}, {4,5}}

%e   5. {{1,2}, {5,6}}

%e   6. {{1,3}, {5,6}}

%e   7. {{1,5}, {2,6}}

%e   8. {{1,5}, {3,6}}

%e   9. {{1,5}, {6,7}}

%e   10. {{5,6}, {7,8}}

%e Likewise, in the 2 X 2 matrix interpretation, a representative from each of the a(2)=10 equivalence classes are

%e   [2 0 ; 0 2]

%e   [1 1 ; 1 1]

%e   [2 0 ; 0 1]

%e   [1 1 ; 1 0]

%e   [2 0 ; 0 0]

%e   [1 1 ; 0 0]

%e   [1 0 ; 1 0]

%e   [1 0 ; 0 1]

%e   [1 0 ; 0 0]

%e   [0 0 ; 0 0]

%t 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 *)

%t LinearRecurrence[{3,-1,-5,5,1,-3,1},{1,3,10,20,43,75,132},60] (* _Harvey P. Dale_, May 28 2021 *)

%o (PARI) 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

%Y Cf. A316587.

%K nonn,easy

%O 0,2

%A _Yu Hin Au_, Jul 24 2020