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Number of permutations of [n] avoiding {4231, 3412, 1234}.
1

%I #12 Nov 10 2017 15:03:53

%S 1,1,2,6,21,71,212,554,1289,2725,5326,9758,16941,28107,44864,69266,

%T 103889,151913,217210,304438,419141,567855,758220,999098,1300697,

%U 1674701,2134406,2694862,3373021,4187891,5160696,6315042,7677089,9275729,11142770,13313126,15825013,18720151,22043972

%N Number of permutations of [n] avoiding {4231, 3412, 1234}.

%H Colin Barker, <a href="/A294725/b294725.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Callan, T. Mansour, <a href="http://arxiv.org/abs/1705.00933">Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns</a>, arXiv:1705.00933 [math.CO] (2017), Table 2 No 24.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F From _Colin Barker_, Nov 10 2017: (Start)

%F G.f.: (1 - 6*x + 16*x^2 - 22*x^3 + 21*x^4 - 6*x^5 + 2*x^6) / (1 - x)^7.

%F a(n) = (1/120)*(120 - 40*n + 44*n^2 - 15*n^3 + 15*n^4 - 5*n^5 + n^6).

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.

%F (End)

%p cn := [1,-6,16,-22,21,-6,2] ;

%p p := add(cn[i]*x^(i-1),i=1..nops(cn)) ;

%p q := (1-x)^7 ;

%p taylor(p/q,x=0,40) ;

%p gfun[seriestolist](%) ;

%o (PARI) Vec((1 - 6*x + 16*x^2 - 22*x^3 + 21*x^4 - 6*x^5 + 2*x^6) / (1 - x)^7 + O(x^40)) \\ _Colin Barker_, Nov 10 2017

%K nonn,easy

%O 0,3

%A _R. J. Mathar_, Nov 07 2017