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 A116731 Number of permutations of length n which avoid the patterns 321, 2143, 3124; or avoid the patterns 132, 2314, 4312, etc. 6
 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300, 2681, 3102, 3565, 4072, 4625, 5226, 5877, 6580, 7337, 8150, 9021, 9952, 10945, 12002, 13125, 14316, 15577, 16910, 18317, 19800, 21361, 23002, 24725, 26532 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums of triangle A130154. Also, binomial transform of [1, 1, 2, 2, 0, 0, 0,...]. - Gary W. Adamson, Oct 23 2007 Conjecture: also counts the distinct pairs (flips, iterations) that a bubble sort program generates when sorting all permutations of 1..n. - Wouter Meeussen, Dec 13 2008 A006527+1=A116731. - Vladimir Joseph Stephan Orlovsky, May 04 2011 a(n) is the number of lattice points (x,y) in the closed region bounded by the parabolas y = x(x - n) and y = x(n - x). - Clark Kimberling, Jun 01 2013 LINKS Guo-Niu Han, Enumeration of Standard Puzzles Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy] Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274) Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005. Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1) FORMULA G.f.: A(x) = {(3x^2-2x+1)x}/{(x-1)^4} a(n) = (n^3-3*n^2+5*n)/3. - Franklin T. Adams-Watters, Sep 13 2006 MATHEMATICA Table[(n^3-3*n^2+5*n)/3, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *) CROSSREFS Cf. A006527, A130154. Sequence in context: A096584 A002836 A116720 * A116722 A116730 A240847 Adjacent sequences:  A116728 A116729 A116730 * A116732 A116733 A116734 KEYWORD nonn,easy AUTHOR Lara Pudwell, Feb 26 2006 EXTENSIONS More terms from Franklin T. Adams-Watters, Sep 13 2006 STATUS approved

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