A289472 Number of gcds-sortable two-rooted graphs on n vertices.

0, 1, 1, 17, 113, 7729, 224689, 61562033, 7309130417, 8013328398001, 3825133597372081, 16776170217003753137, 32072986971771549318833, 562672074981014060438175409, 4304275145962667488546071527089, 302049699050029408242290021253725873

Offset

1,4

Comments

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 4^rank(A) two-rooted gcds-sortable graphs containing the non-root subgraph G. 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.

Links

Table of n, a(n) for n=1..16.

C. A. Brown, C. S. Carrillo Vazquez, R. Goswami, S. Heil, and M. Scheepers, The Sortability of Graphs and Matrices Under Context Directed Swaps

F. J. MacWilliams, Orthogonal matrices over finite fields, Amer. Math. Monthly, 76 (1969), 152-164.

Formula

a(n) = Sum_{s=0..floor(n/2)-1} 2^(s^2+3s) * (Product_{i=0..2s-1} (2^(n-2-i)-1) / Product_{i=1..s} (2^(2i)-1)).

Mathematica

Table[Sum[2^(s^2 + 3 s) (Product[(2^(n - 2 - i) - 1), {i, 0, 2 s - 1}]/Product[(2^(2 i) - 1), {i, s}]), {s, 0, Floor[n/2] - 1}], {n, 16}] (* Michael De Vlieger, Jul 30 2017 *)

Prog

(PARI) a(n) = sum(s=0, n\2-1, 2^(s^2+3*s)*prod(i=0, 2*s-1, (2^(n-2-i)-1))/prod(i=1, s, 2^(2*i)-1)); \\ Michel Marcus, Jul 07 2017

Crossrefs

Sequence in context: A299223 A197360 A367940 * A172521 A349879 A226688

Adjacent sequences: A289469 A289470 A289471 * A289473 A289474 A289475

Keyword

nonn

Author

Sam Heil, Jul 06 2017

Status

approved