A288965 Number of key comparisons to sort all n! permutations of n elements by the optimal dual-pivot quicksort.

0, 0, 2, 16, 114, 866, 7188, 65580, 655872, 7157376, 84775680, 1084343040, 14906039040, 219267751680, 3437854963200, 57247256424960, 1009189972869120, 18779054120386560, 367876307230064640, 7568437652936294400, 163164173556503347200, 3678547646214424166400

Offset

0,3

Links

Daniel Krenn, Table of n, a(n) for n = 0..100

M. Aumüller, M. Dietzfelbinger, C. Heuberger, D. Krenn, and H. Prodinger, Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths, arXiv:1611.00258 [math.CO], 2016.

M. Aumüller, M. Dietzfelbinger, C. Heuberger, D. Krenn, and H. Prodinger, Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths, Combin. Probab. Comput. 28(4) (2019), 485-518.

Daniel Krenn, Quickstar, Program in SageMath, on GitHub.

R. Neininger and J. Straub, Probabilistic analysis of the dual-pivot quicksort "count", arXiv:1710.07505 [cs.DS], 2017.

R. Neininger and J. Straub, Probabilistic analysis of the dual-pivot quicksort "count", 2018 Proceedings of the Meeting of Analytic Algorithmics and Combinatorics, New Orleans, Louisiana, USA, 7pp.

Index entries for sequences related to sorting

Formula

a(n) = n!*((9/5)*n*Harmonic(n) - (1/5)*n*Harmonic_alt(n) - (89/25)*n + (67/40)*Harmonic(n) - (3/40)*Harmonic_alt(n) - 83/800 + (-1)^n/10 - ([n even]/320)*(1/(n - 3) + 3/(n - 1)) + ([n odd]/320)*(3/(n - 2) + 1/n)) for n >= 4, where [condition] = 1 if the condition holds, and 0 otherwise, and Harmonic_alt(n) = Sum_{k=1..n} (-1)^n/n = -A058313(n)/A058312(n). [This follows from Theorem 12.1 in Aumüller et al. (2016) or Theorem 5.7 in Aumüller et al. (2019).] - Petros Hadjicostas, May 03 2020

Maple

haralt := proc(n) local k: add((-1)^k/k, k = 1 .. n): end proc:
a := proc(n) local v, v1, v2: if n = 0 or n = 1 then v := 0: end if; if n = 2 then v := 2: end if: if n = 3 then v := 16: end if:
if 4 <= n then v1 := 9/5*n*harmonic(n) - 1/5*n*haralt(n) - 89/25*n + 67/40*harmonic(n) - 3/40*haralt(n) - 83/800 + 1/10*(-1)^n:
if 0 = n mod 2 then v2 := -1/320*1/(n - 3) - 3/320*1/(n - 1): end if:
if 1 = n mod 2 then v2 := 3/320*1/(n - 2) + 1/320*1/n: end if:
v := n!*(v1 + v2): end if: v: end proc:
seq(a(n1), n1 = 0 .. 30); # Petros Hadjicostas, May 03 2020

Mathematica

haralt[n_] := Sum[(-1)^k/k, {k, 1, n}];
a[n_] := Switch[n, 0|1, 0, 2, 2, 3, 16, _, n! ((9/5) n HarmonicNumber[n] - (1/5) n haralt[n] - (89/25) n + (67/40) HarmonicNumber[n] - (3/40)* haralt[n] - 83/800 + (-1)^n/10 - (Boole[EvenQ[n]]/320)(1/(n-3) + 3/(n-1)) + (Boole[OddQ[n]]/320)(3/(n-2) + 1/n))];
a /@ Range[0, 21] (* Jean-François Alcover, Jun 05 2020 *)

Prog

(PARI) lista(nn) = {my(x='x + O('x^nn)); concat([0, 0], Vec(serlaplace(-8*log(1-x)/(5*(1-x)^2) + 2*atanh(x)/(5*(1-x)^2) - 44/(25*(1-x)^2) - atanh(x)/(4*(1-x)) + 281/(160*(1-x)) + ((1-x)^3/320)*atanh(x) + x^3/150 - 27*x^2/1600 + 17*x/1600 + 3/800)))}; \\ Petros Hadjicostas and Michel Marcus, May 04 2020, e.g.f. from p. 29 in Aumüller et al. (2016)

Crossrefs

Cf. A058312, A058313, A288964, A288970, A288971.

Sequence in context: A037564 A361743 A125725 * A207420 A207736 A208364

Adjacent sequences: A288962 A288963 A288964 * A288966 A288967 A288968

Keyword

nonn

Author

Daniel Krenn, Jun 20 2017

Status

approved