A113869 Coefficients in asymptotic expansion of probability that a random pair of elements from the alternating group A_k generates all of A_k.

1, -1, -1, -4, -23, -171, -1542, -16241, -194973, -2622610, -39027573, -636225591, -11272598680, -215668335091, -4431191311809, -97316894892644, -2275184746472827, -56421527472282127, -1479397224086870294, -40897073524132164189, -1188896226524012279617

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

L. Babai, The probability of generating the symmetric group, J. Combin. Theory, A52 (1989), 148-153.

J. Bovey and A. Williamson, The probability of generating the symmetric group, Bull. London Math. Soc. 10 (1978) 91-96.

J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969) 199-205.

J. D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.

Thibault Godin, An analogue to Dixon's theorem for automaton groups, arXiv preprint arXiv:1610.03301 [math.GR], 2016.

Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Formula

The probability that a random pair of elements from the alternating group A_k generates all of A_k is P_k ~ 1-1/k-1/k^2-4/k^3-23/k^4-171/k^5-... = Sum_{n >= 0} a(n)/k^n.

Furthermore, P_k ~ 1 - Sum_{n >= 1} A003319(n)/[k]_n, where [k]_n = k(k-1)(k-2)...(k-n+1). Therefore for n >= 2, a(n) = - Sum_{i=1..n} A003319(i)*Stirling_2(n-1, i-1). - N. J. A. Sloane.

a(n) ~ -n! / (4 * (log(2))^(n+1)). - Vaclav Kotesovec, Jul 28 2015

Mathematica

A003319[n_] := A003319[n] = n! - Sum[ k!*A003319[n-k], {k, 1, n-1}]; a[n_] := -Sum[ A003319[i]*StirlingS2[n-1, i-1], {i, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 11 2012, after N. J. A. Sloane *)

Crossrefs

Cf. A003319, A112354, A113871, A114038, A158094, A168246, A306153, A316084.

Sequence in context: A277382 A208676 A317276 * A084357 A360868 A075729

Adjacent sequences: A113866 A113867 A113868 * A113870 A113871 A113872

Keyword

sign,nice

Author

N. J. A. Sloane, Jan 26 2006

Status

approved