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%I #56 Feb 06 2023 09:36:17
%S 1,2,5,12,25,46,77,120,177,250,341,452,585,742,925,1136,1377,1650,
%T 1957,2300,2681,3102,3565,4072,4625,5226,5877,6580,7337,8150,9021,
%U 9952,10945,12002,13125,14316,15577,16910,18317,19800,21361,23002,24725,26532
%N Number of permutations of length n which avoid the patterns 321, 2143, 3124; or avoid the patterns 132, 2314, 4312, etc.
%C Row sums of triangle A130154. Also, binomial transform of [1, 1, 2, 2, 0, 0, 0, ...]. - _Gary W. Adamson_, Oct 23 2007
%C 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
%C 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
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, <a href="http://arxiv.org/abs/1302.2274">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, arXiv:1302.2274 [math.CO], 2013.
%H Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, <a href="https://www.emis.de/journals/INTEGERS/papers/p16/p16.Abstract.html">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Integers: Electronic Journal of Combinatorial Number Theory, 15 (2015), #A16.
%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/maple/webbook/bookmain.html">Systematic Studies in Pattern Avoidance</a>, 2005.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1)
%F G.f.: (3*x^2 - 2*x + 1)*x/(x - 1)^4.
%F a(n) = (n^3 - 3*n^2 + 5*n)/3. - _Franklin T. Adams-Watters_, Sep 13 2006
%F a(n) = A006527(n-1) + 1. - _Vladimir Joseph Stephan Orlovsky_, May 04 2011
%F E.g.f.: exp(x)*(x + x^3/3). - _Nikolaos Pantelidis_, Feb 05 2023
%t Table[(n^3-3*n^2+5*n)/3,{n,100}] (* _Vladimir Joseph Stephan Orlovsky_, May 04 2011 *)
%Y Cf. A006527, A130154.
%K nonn,easy
%O 1,2
%A _Lara Pudwell_, Feb 26 2006
%E More terms from _Franklin T. Adams-Watters_, Sep 13 2006
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