A271214 Number of reduced rearrangement patterns with n blocks.

1, 1, 2, 10, 71, 653, 7638, 104958, 1664083, 29740057, 591645738, 12959409010, 309898317151, 8032551265957, 224316415082750, 6714021923017318, 214415538303362411, 7277133405318569009, 261560966377901961810, 9925178291099012783322, 396498148141095399675511

Offset: 0

Author

Jonathan Burns, Apr 13 2016

Keywords

• nonn
• easy

Comments

a(n) is the number of reduced rearrangement patterns, i.e., the number of reduced rearrangement map equivalence classes formed from the two rotation involutions.

Examples

			For n=0 the a(0)=1 solution is { ∅ }
For n=1 the a(1)=1 solution is { +1 }
For n=2 the a(2)=2 solutions are { +2+1, +1-2 }
For n=3 the a(3)=10 solutions are { +3-2+1, +1+3-2, +2-3+1, +1+3+2, +2+1-3, +3+1-2, +1-3+2, +3+2+1, +3+2-1, +1-2+3 }
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• J. Burns, Table of Rearrangement Maps and Patterns for n = 1, 2, and 3.

Crossrefs

Cf. A271212, A271217.

Programs

Mathematica

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

Formula

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

a(n) ~ sqrt( Pi*n / 8*e) * (2n / e)^n.

a(n) = (A271212(n) + A271217(n)) / 4.