A093418 Numerator of -3*n + 2*(1+n)*HarmonicNumber(n).

0, 1, 3, 17, 53, 62, 163, 717, 3489, 3727, 43391, 45596, 619313, 644063, 667229, 2756003, 24124223, 24784883, 160941559, 164719333, 33664415, 11451017, 268428987, 819836496, 20845424563, 21181779967, 193553388003, 196368607553, 5773568883787, 5849866645327

Offset: 0

Author

Amarnath Murthy, Mar 30 2004

Keywords

• nonn
• frac

Comments

Numerator of the average time to quicksort n items in random order.

From Petros Hadjicostas, Oct 20 2019: (Start)

Depending on the assumptions used in the literature, the average number to sort n items in random order by quicksort appears as -C*n + 2*(1+n)*HarmonicNumber(n), where C = 2, 3, or 4. See, for example, A115107 and A288964. Other variations of the above formula are possible.

Let X_n be the random number of comparisons needed to sort a list of numbers in the symmetric group S_n and let R_n be the rank of the pivot. According to Havil (2003, pp. 128-130), we have X_n = n + X_{R_n-1} + X_{n-R_n} because it takes 1 unit of comparison time to pick the pivot and n-1 comparisons to divide the data into two lists of numbers (less than the pivot and greater than the pivot). No matter how we pick the pivot, we have to assume that R_n is jointly independent of (X_1, ..., X_n). We let X_0 = 0.

Denoting expectation by E(.) and conditional expectation by E(.|.), we have E(X_n) = Sum_{r = 1..n} E(n + X_{R_n-1} + X_{n-R_n} | R_n=r) * P(R_n=r) = n + (1/n) * (E(X_{r-1}) + E(X_{n-r})) The last step follows from the assumed independence of R_n from (X_1, ..., X_n). This simplifies to E(X_n) = n + (2/n) * Sum_{r = 0..n-1} E(X_r). As in Havil (2003), solving the recurrence we get E(X_n) = fr(n) = -3*n + 2*(1+n)*HarmonicNumber(n). Here a(n) = numerator(E(X_n)) and A096620(n) = denominator(E(X_n)).

Note that E(X_n)*n! = (-3*n + 2*(1+n)*HarmonicNumber(n)) * n! = A063090(n), and according to the documentation of that sequence, A063090(n)/(n*n!) is the "average number of comparisons needed to find a node in a binary search tree containing n nodes inserted in a random order". See Knuth (1998, Vol. 3, pp. 430-431 and Exercise 6 on pp. 454-455).

Other authors (e.g., Cameron (1996)) do not count the choice of the pivot as a comparison time. In such a case, if we let Y_n be the modified number of comparisons used by quicksort to sort a random list of length n, we get the modified recurrence Y_n = n - 1 + Y_{R_n-1} + Y_{n-R_n}, from which we get E(Y_n) = n - 1 + (2/n) * Sum_{r = 0..n-1} E(Y_r). Solving this modified recurrence, we get E(Y_n) = -4*n + + 2*(1+n)*HarmonicNumber(n). In such a case, A115107(n) = numerator(E(Y_n)) = numerator(-4*n + 2*(1+n)*HarmonicNumber(n)) and A288964(n) = n! * E(Y_n) = n! * (-4*n + 2*(1+n)*HarmonicNumber(n)).

(End)

Examples

			fr(n) = 0, 1, 3, 17/3, 53/6, 62/5, 163/10, 717/35, 3489/140, ... = A093418/A096620.
		

References

• Peter J. Cameron, Combinatorics: Topics, Techniques and Algorithms, Cambridge Univ. Press, 1996; see pp. 66-68.
• J. H. Conway and R. K. Guy, The Book of Numbers. New York: Springer-Verlag, 1996, pp. 143 and 258-259.
• Julian Havil, Gamma: Exploring Euler's constant, Princeton University Press, 2003; see pp. 128-130.
• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, 1998; see pp. 427-431 and 454-455.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 [recomputed for offset 0 by _Georg Fischer_, Oct 13 2019]
• S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019. [They have the recurrence with n-1, rather than n, from which they get -4*n + 2*(1+n)*HarmonicNumber(n) for the average number of comparisons.]
• M. Kauers and P. Paule, The Concrete Tetrahedron, Springer, 2011; see p. 4. [They agree with Cameron's recurrence and the numerators are in A115107.]
• Eric Weisstein's World of Mathematics, Quicksort. [He uses Havil's recurrence.]
• Eric Weisstein's World of Mathematics, Harmonic Number.
• Wikipedia, Quicksort. [Uses Cameron's recurrence and the numerators are in A115107.]

Crossrefs

Cf. A001008, A002805, A063090, A096620, A115107, A288964.

The denominators in A096620 are essentially the same as the numbers in A093419, which are the denominators of A093412.

Programs

Magma

[Numerator(2*(n+1)*HarmonicNumber(n) -3*n): n in [0..50]]; // G. C. Greubel, Sep 01 2018

Maple

a := n -> numer(2*(n + 1)*(Psi(n + 1) + gamma) - 3*n);
seq(a(n), n=0..26); # Peter Luschny, Aug 26 2019

Mathematica

Numerator[Table[2*Sum[Sum[1/i, {i, 1, k}], {k, 1, n}]-n, {n, 0, 20}]] (* or *) Numerator[Table[2*Sum[HarmonicNumber[k], {k, 1, n}]-n, {n, 0, 20}]]
Numerator[Table[2*(n+1)*HarmonicNumber[n] - 3*n, {n, 0, 50}]] (* G. C. Greubel, Sep 01 2018 *)

PARI

{h(n) = sum(k=1,n,1/k)};
for(n=0,50, print1(numerator(2*(n+1)*h(n) -3*n), ", ")) \\ G. C. Greubel, Sep 01 2018

Python

from fractions import Fraction
from itertools import count, islice
def agen(): # generator of terms
    Hn = Fraction(0, 1)
    for n in count(0):
        yield (-3*n + 2*(1+n)*Hn).numerator
        Hn += Fraction(1, n+1)
print(list(islice(agen(), 27))) # Michael S. Branicky, Apr 17 2022

Formula

G.f. for fractions fr(n): -(x+2*log(1-x))/(1-x)^2. - Vladeta Jovovic, Jul 05 2004

a(n) = 1 + (1/n!) * Sum_{k=0..n} (-1)^(n-k) * Stirling1(n, k) * (k-1) * 2^k. - Vladeta Jovovic, Jul 05 2004

a(n) = numerator(-n + 2 * H^{2}(n)), where H^{2}(n) = Sum_{k=1..n} HarmonicNumber(k) is second-order harmonic number. - Alexander Adamchuk, Nov 01 2004

Extensions

Edited by Eric W. Weisstein, Jul 01 2004

Offset changed to 0 by Petros Hadjicostas, Aug 26 2019