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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. 7
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

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

Table of n, a(n) for n=1..44.

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, arXiv:1302.2274 [math.CO], 2013.

Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Integers: Electronic Journal of Combinatorial Number Theory, 15 (2015), #A16.

Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)

FORMULA

G.f.: (3*x^2 - 2*x + 1)*x/(x - 1)^4.

a(n) = (n^3 - 3*n^2 + 5*n)/3. - Franklin T. Adams-Watters, Sep 13 2006

a(n) = A006527(n-1) + 1. - Vladimir Joseph Stephan Orlovsky, May 04 2011

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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Last modified December 9 19:36 EST 2022. Contains 358703 sequences. (Running on oeis4.)