A271217 Number of symmetric reduced rearrangement maps.

1, 2, 2, 6, 22, 50, 274, 598, 4486, 9570, 90914, 191398, 2201078, 4593554, 62012978, 128619510, 1993602406, 4115824322, 72026925634, 148169675590, 2889308674006

Offset: 0

Author

Jonathan Burns, Apr 13 2016

Keywords

• nonn
• easy

Comments

a(n) is the number of reduced rearrangement maps on n blocks. A rearrangement map is a signed permutation, e.g., +2 -1 -3. If the permutation contains (i)(i+1) or -(i+1)-(i) for any i, then it is not reduced. The map a is symmetric if a=a^(AI) and a^A = a^I where A and I are the rotation involutions.

Examples

			For n=0 the a(0)=1 solution is { ∅ }
For n=1 the a(1)=2 solutions are { +1, -1 }
For n=2 the a(2)=2 solutions are { +2+1, -1-2 }
For n=3 the a(3)=6 solutions are { +3-2+1, -1+2-3, +3+2+1, -1-2-3, +1-2+3, -3+2-1 }
		

References

• J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

Links

• J. Burns, Table of Rearrangement Maps and Patterns for n = 1, 2, and 3.

Crossrefs

A271217 / A271216 ~ e^(-1/4).

Cf. A271212, A271214

Programs

Mathematica

Table[Round[2^n*Exp[-1/4]*(1-(1+(-1)^n)/(4 n))*Floor[n/2]!],{n,1,20}]

Formula

a(n) = round( 2^n * e^(-1/4) * ( 1 - (1 + (-1)^n)/(4n) ) * floor(n/2)! )

a(2k+1) = 2*a(2k) + a(2k-1) and a(2k) = (2k-1)*a(2k-1)+(2k-2)*a(2k-3)

a(n) ~ e^(-1/4) * 2^n * floor(n/2)!.

Conjecture: (-2*n+9)*a(n) -4*a(n-1) +(2*n-3)*(2*n-7)*a(n-2) -4*a(n-3) +2*(2*n-5)*(n-4)*a(n-4)=0. - R. J. Mathar, Jan 04 2017