|
%I #22 Jun 03 2018 02:01:09
%S 0,1,1,5,29,365,7565,259533,16766541,1695913805,319025518925,
%T 99428910374221,53629954918196557,51436455420773021005,
%U 81633965668282476025165,234346782219278654389392717,1131832076434284133556933170509
%N Number of gcds-sortable two-rooted graphs on n vertices such that all vertices have even degree.
%C This formula comes from the fact that for each possible value of the (n-2)-vertex subgraph G containing all of the non-root vertices, if G has adjacency matrix A over F_2 then there are 2^rank(A) two-rooted gcds-sortable graphs with all vertices of even degree containing the non-root subgraph G. Then, we can apply the formula from MacWilliams for the number of symmetric binary matrices with zero diagonal of each rank to get the total number of gcds-sortable graphs with all vertices of even degree.
%H C. A. Brown, C. S. Carrillo Vazquez, R. Goswami, S. Heil, and M. Scheepers, <a href="http://diamond.boisestate.edu/reu/publications/2017CDS.pdf">The Sortability of Graphs and Matrices Under Context Directed Swaps</a>
%H F. J. MacWilliams, <a href="http://www.jstor.org/stable/2317262">Orthogonal matrices over finite fields</a>, Amer. Math. Monthly, 76 (1969), 152-164.
%F a(n) = Sum_{s=0..floor(n/2)-1} 2^((s^2+3s)/2) * (Product_{i=0..2s-1} (2^(n-2-i)-1) / Product_{i=1..s} (2^(2i)-1))
%t Table[Sum[2^((s^2 + 3 s)/2) * Product[(2^(n - 2 - i) - 1), {i, 0, 2 s - 1}]/Product[(2^(2 j) - 1), {j, s}], {s, 0, Floor[n/2] - 1}], {n, 2, 17}] (* _Michael De Vlieger_, Jul 12 2017 *)
%o (PARI) a(n) = sum(s=0, n\2-1, 2^((s^2+3*s)/2)*prod(i=0, 2*s-1, (2^(n-2-i)-1))/prod(i=1, s, 2^(2*i)-1)); \\ _Michel Marcus_, Jul 07 2017
%Y Cf. A289472.
%K nonn
%O 1,4
%A _Sam Heil_, Jul 06 2017
|
