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A260509
Number of graphs on labeled vertices {x, y, 1, 2, ..., n}, such that there is a partition of the vertices into V_1 and V_2 with x in V_1, y in V_2, every v in V_1 adjacent to an even number of vertices in V_2, and every v in V_2 adjacent to an even number of vertices in V_1.
1
1, 3, 23, 351, 11119, 703887, 89872847, 22945886799, 11740984910671, 12014755220129103, 24602393557227030863, 100754627840184914661711, 825349838279823049359417679, 13521969078301639826644261077327, 443083578482642171171990600910324047, 29037623349739387300519333731237743018319
OFFSET
0,2
COMMENTS
a(n) is also the number of graphs on vertices {x, y, 1, 2, ..., n} that can be sorted to the discrete graph by a series of gcdr and even-gcdr moves.
Asymptotically, a(n) is a third of the total number of graphs, i.e., lim_{n->infinity} (a(n) / 2^(binomial(n+2, 2)) = 1/3.
LINKS
FORMULA
a(n) + (2^n - 1)*a(n-1) = 2^(binomial(n+2, 2) - 1) = 2^(n^3 + 3n).
a(n) = Sum_{k=0..n} (Product_{i=1..k} 2^(i+1))(Product_{i=k+1..n} (1 - 2^i)).
Exponential generating function A(x) satisfies A(0) = 1 and A'(x) + 2A(2x) - A(x) = 4F(8x). Here F(x) is the exponential generating function counting the graphs on n labeled vertices, and satisfies F(0) = 1 and F'(x) = F(2x).
EXAMPLE
a(2) = 23 because there are 23 graphs on {x, y, 1, 2} that admit a vertex partition separating x and y, such that each vertex in one half of the partition is adjacent to an even number of vertices in the other half. For instance, the graph with four edges (x,y), (x,1), (y,2), (1,2) admits the partition {{x,2},{y,1}}.
PROG
(Python3)
# a_1(n) and a_2(n) both count the same sequence, in two different ways.
def a_1(n) :
# Output the number of 2-rooted graphs in (a) with n+2 vertices
if n == 0 :
return 1
else :
return 2**((n*n + 3*n) // 2) - (2**n - 1) * a_1(n-1)
def a_2(n) :
# Output the number of 2-rooted graphs in (a) with n+2 vertices
# Formula: \sum_{k=0}^n (\prod_{i=1}^k 2^{i+1}) (\prod_{i=k+1}^n (1 - 2^i))
curr_sum = 0
for k in range(0, n+1) :
curr_prod = 1
for i in range(1, k+1) :
curr_prod *= (2**(i+1))
for i in range(k+1, n+1) :
curr_prod *= (1 - (2**i))
curr_sum += curr_prod
return curr_sum
(PARI) a(n) = sum(k=0, n, prod(i=1, k, 2^(i+1))*prod(i=k+1, n, 1 - 2^i)); \\ Michel Marcus, Sep 11 2015
CROSSREFS
Cf. A260506 (counts the special case where the graph in question is required to be the overlap graph of some signed permutation).
Cf. A006125 (the total number of graphs on n labeled vertices).
Sequence in context: A377742 A092664 A363577 * A073588 A068338 A255881
KEYWORD
nonn
AUTHOR
Caleb Stanford, Jul 27 2015
STATUS
approved