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A165328 The number of even separable permutations of length n 0
1, 1, 3, 12, 48, 197, 903, 4298, 20862, 103049, 518859, 2647296, 13651092, 71039373, 372693519, 1968822294, 10463661690, 55909013009, 300159426963, 1618362990804, 8759313066840, 47574827600981, 259215937709463 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For n congruent to 2 or 3 mod 4, these are the little Schroeder numbers A001003, because the separable permutations are closed under reversal, and for these values of n, reversing the permutation corresponds to multiplying by an odd permutation. Thus for these values of n, precisely half of the separable permutations are even. For other values of n, it appears that strictly more than half of the separable permutations are even.

LINKS

Table of n, a(n) for n=1..23.

M. Albert, M. D. Atkinson, and V. Vatter, Even separable permutations

FORMULA

G.f. f satisfies 4096*f^12 + (- 24576 + 24576*x)*f^11 + (- 116736*x + 65536 + 65536*x^2)*f^10 + (- 102400 + 235520*x + 102400*x^3 - 235520*x^2)*f^9 + (104000*x^4 - 259584*x + 327040*x^2 + 103744 - 259584*x^3)*f^8 + (163072*x - 70912 - 196096*x^2 + 196096*x^3 + 71936*x^5 - 164096*x^4)*f^7 + (34464*x^6 - 52288*x^5 + 27520*x^3 + 5600*x^2 - 48704*x + 7296*x^4 + 32640)*f^6 + (- 5952*x + 63776*x^2 + 480*x^6 - 9472 - 58688*x^5 + 11360*x^7 - 115040*x^3 + 113536*x^4)*f^5 + (7312*x^7 - 34528*x^6 + 2484*x^8 + 59440*x^3 - 40248*x^2 + 56496*x^5 - 63284*x^4 + 1272 + 11792*x)*f^4 +

+ (- 7344*x^3 + 10800*x^2 - 4848*x - 472*x^4 + 328*x^9 + 2904*x^8 + 152*x^5 + 6656*x^6 - 8320*x^7 + 144)*f^3 + (- 429*x^2 + 882*x + 528*x^9 - 554*x^8 - 2632*x^7 + 20*x^10 - 11750*x^5 + 10471*x^4 - 4484*x^3 + 8045*x^6 - 81)*f^2 + (40*x^10 + 122*x^9 + 9 + 1961*x^7 - 3087*x^6 + 4129*x^5 - 874*x^8 + 2247*x^3 - 513*x^2 - 27*x - 4007*x^4)*f - 351*x^3 - 9*x + 99*x^2 + 615*x^4 - 78*x^9 - 603*x^5 + 361*x^6 - 183*x^7 + 130*x^8 + 20*x^10 = 0.

EXAMPLE

For n=4 there are 22 separable permutations, and 12 of these are even. Thus a(4)=12.

CROSSREFS

Cf. A001003

Sequence in context: A323261 A103943 A283679 * A142873 A301578 A151168

Adjacent sequences:  A165325 A165326 A165327 * A165329 A165330 A165331

KEYWORD

nonn

AUTHOR

Vincent Vatter, Sep 15 2009

STATUS

approved

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Last modified May 19 06:48 EDT 2019. Contains 323386 sequences. (Running on oeis4.)