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A165522
The number of 54321-avoiding separable permutations of length n.
3
1, 1, 2, 6, 22, 89, 368, 1488, 5831, 22311, 84223, 316181, 1185884, 4452567, 16742230, 63025805, 237423928, 894681874, 3371727204, 12706639594, 47884046357, 180440982667, 679939553548, 2562134671440, 9654584875285, 36380338185856, 137088669193146
OFFSET
0,3
LINKS
V. Vatter, Finding regular insertion encodings for permutation classes, Journal of Symbolic Computation, Volume 47, Issue 3, March 2012, Pages 259-265.
Index entries for linear recurrences with constant coefficients, signature (18, -148, 743, -2564, 6488, -12536, 18999, -22992, 22474, -17876, 11622, -6189, 2697, -957, 273, -61, 10, -1).
FORMULA
G.f.: (1-x)^4*(1-3*x+2*x^2-x^3)^2*(1-7*x+19*x^2-28*x^3+23*x^4 -12*x^5 +4*x^6-x^7) / (x^18 -10*x^17 +61*x^16 -273*x^15 +957*x^14 -2697*x^13 +6189*x^12 -11622*x^11 +17876*x^10 -22474*x^9 +22992*x^8 -18999*x^7 +12536*x^6 -6488*x^5 +2564*x^4 -743*x^3 +148*x^2 -18*x +1). [typo fixed by Colin Barker, Jul 05 2013]
The growth rate (limit of the n-th root of a(n)) is approximately 3.76823.
EXAMPLE
For n=6, there are 394 separable permutations; 368 of them avoid 54321.
MATHEMATICA
CoefficientList[Series[(1-x)^4*(1-3*x+2*x^2-x^3)^2*(1-7*x+19*x^2-28*x^3 + 23*x^4-12*x^5+4*x^6-x^7)/(x^18-10*x^17+61*x^16-273*x^15+957*x^14- 2697*x^13+6189*x^12-11622*x^11+17876*x^10-22474*x^9+22992*x^8-18999*x^7 +12536*x^6-6488*x^5+2564*x^4-743*x^3+148*x^2-18*x+1), {x, 0, 50}], x] (* G. C. Greubel, Oct 21 2018 *)
PROG
(PARI) x='x+O('x^50); Vec((1-x)^4*(1-3*x+2*x^2-x^3)^2*(1-7*x+19*x^2 -28*x^3+23*x^4-12*x^5+4*x^6-x^7)/(x^18-10*x^17+61*x^16 -273*x^15 +957*x^14 -2697*x^13+6189*x^12-11622*x^11+17876*x^10-22474*x^9 +22992*x^8 -18999*x^7+12536*x^6-6488*x^5+2564*x^4-743*x^3+148*x^2 -18*x+1)) \\ G. C. Greubel, Oct 21 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)^4*(1-3*x+2*x^2-x^3)^2*(1-7*x+19*x^2-28*x^3 + 23*x^4-12*x^5+4*x^6-x^7)/(x^18-10*x^17+61*x^16-273*x^15+957*x^14- 2697*x^13+6189*x^12 -11622*x^11+17876*x^10-22474*x^9+22992*x^8-18999*x^7 +12536*x^6-6488*x^5 +2564*x^4-743*x^3+148*x^2-18*x+1))); // G. C. Greubel, Oct 21 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincent Vatter, Sep 21 2009
STATUS
approved