A116731 Number of permutations of length n which avoid the patterns 321, 2143, 3124; or avoid the patterns 132, 2314, 4312, etc.

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

Offset: 1

Author

Lara Pudwell, Feb 26 2006

Keywords

• nonn
• easy

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

• 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)

Crossrefs

Cf. A006527, A130154.

Programs

Mathematica

Table[(n^3-3*n^2+5*n)/3,{n,100}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)

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

E.g.f.: exp(x)*(x + x^3/3). - Nikolaos Pantelidis, Feb 05 2023

Extensions

More terms from Franklin T. Adams-Watters, Sep 13 2006