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 A271214 Number of reduced rearrangement patterns with n blocks. 3
 1, 1, 2, 10, 71, 653, 7638, 104958, 1664083, 29740057, 591645738, 12959409010, 309898317151, 8032551265957, 224316415082750, 6714021923017318, 214415538303362411, 7277133405318569009, 261560966377901961810, 9925178291099012783322, 396498148141095399675511 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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. 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. 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. EXAMPLE 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 } 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}] CROSSREFS Cf. A271212, A271217. Sequence in context: A362821 A060842 A245834 * A366241 A321446 A111554 Adjacent sequences: A271211 A271212 A271213 * A271215 A271216 A271217 KEYWORD nonn,easy AUTHOR Jonathan Burns, Apr 13 2016 STATUS approved

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Last modified May 18 03:12 EDT 2024. Contains 372617 sequences. (Running on oeis4.)