1,3

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.

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

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.

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

Cf. A001003

Sequence in context: A164346 A113956 A103943 this_sequence A142873 A151168 A151169

Adjacent sequences: A165325 A165326 A165327 this_sequence A165329 A165330 A165331

nonn

Vince Vatter (vatter(AT)gmail.com), Sep 15 2009