A288970 -id:A288970 - cp's OEIS frontend

A335990 The moment generating function of the limiting distribution of the number of comparisons in quicksort can be written in the form M(t) = m(-2t)/(exp(2gammat)Gamma(1 + 2t)) for |t| < 1/2, where m(z) = Sum_{n >= 0} B(n)z^n/n! for |z| < 1. This sequence gives the numerators of the rational numbers B(n) for n >= 0.

1, 0, 7, 19, 565, 229621, 74250517, 30532750703, 90558126238639, 37973078754146051, 21284764359226368337, 1770024989560214080011109, 539780360793818428471498394131, 194520883210026428577888559667954807, 911287963487139630688627952818633149408727, 328394760901508739430228985010652235796369497219

Offset: 0

Petros Hadjicostas, Jul 03 2020

Despite the fact that both the numerator and denominator in the formula M(t) = m(-2*t)/(exp(2*gamma*t)*Gamma(1 + 2*t)) each have a Taylor expansion around t = 0 with a radius of convergence equal to 1/2, the moment generating function M(t) has a Taylor expansion around t = 0 with an infinite radius of convergence. This was proved in Rösler (1991).
The formula for M(t) appears as Theorem 6.1 in Tan and Hadjicostas (1993) and derives from the work of Hennequin (1991). Hennequin conjectured a cumulant formula for the limiting distribution of the number of comparisons in quicksort in his 1989 paper, and he proved it in his 1991 thesis.
The numbers (B(n): n >= 0), with B(0) = 1 and B(0) = 0, are given (for p >= 0) by the recurrence.
Sum_{r=0..p} Stirling1(p+2, r+1)*B(p-r)/(p-r)! + Sum_{r=0..p} F(r)*F(p-r) = 0, where F(r) = Sum_{i=0..r} Stirling1(r+1,i+1)*G(r-i) and G(k) = Sum_{a=0..k} (-1)^a*B(k-a)/(a!*(k-a)!*2^a).
The numbers A(n) = L_n(B(1),...,B(n)) = A330852(n)/A330860(n), where L_n(x_1,...,x_n) are the logarithmic polynomials of Bell, appear in Hennequin's cumulant formula.
Hoffman and Kuba (2019, 2020) gave an alternative proof of Hennequin's cumulant formula and gave an alternative calculation for the constants (-2)^n*A(n), which they denote by a_n. See also Finch (2020).
Hoffman and Kuba (2019-2020, Proposition 17) express the constants c(n) = B(n)*(-2)^n = A329001(n)/A330876(n) in terms of "tiered binomial coefficients". In terms of the constants c(n), the moment generating function equals M(t) = Sum_{n >= 0} (c(n)*t^n/n!)/(exp(2*gamma*t)*Gamma(1 + 2*t)) for |t| < 1/2.
Tan and Hadjicostas (1993) proved that lim_{n -> infinity} B(n)/n! = nu, where nu = 0.589164... (approximately). Also, M(-1/2) = nu*exp(gamma), where gamma = A001620 (Euler's constant). (It seems that nu is close to Pi^(1/3) * exp(-1/3 - gamma), but we have no theoretical evidence for that.)
The PARI program below is based on a Maple program in Tan and Hadjicostas (1993).
The rest of the references give the theory of the limiting distribution of the number of comparisons in quicksort (and for that reason we omit any discussion of that topic).

			The first few fractions are 1/1, 0/1, 7/4, 19/8, 565/36, 229621/3456, 74250517/172800, 30532750703/10368000, 90558126238639/3810240000, ... = A335990/A335991.

Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991), p. 83.

S. B. Ekhad and D. Zeilberger, A detailed analysis of quicksort running time, arXiv:1903.03708 [math.PR], 2019. [They have the first eight moments for the number of comparisons in quicksort from which Hennequin's first eight asymptotic cumulants can be derived.]
James A. Fill and Svante Janson, Smoothness and decay properties of the limiting Quicksort density function, In: D. Gardy and A. Mokkadem (eds), Mathematics and Computer Science, Trends in Mathematics, Birkhäuser, Basel, 2000, pp. 53-64.
James A. Fill and Svante Janson, Quicksort asymptotics, Journal of Algorithms, 44(1) (2002), 4-28.
Steven Finch, Recursive PGFs for BSTs and DSTs, arXiv:2002.02809 [cs.DS], 2020; see Section 1.4. [He gives the constants a_s = (-2)^s*A(s) for s >= 2. He also evaluates c(2) - c(8), where c(n) = B(n)*(-2)^n.]
P. Hennequin, Combinatorial analysis of the quicksort algorithm, Informatique théoretique et applications, 23(3) (1989), 317-333.
M. E. Hoffman and M. Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020; see Section 5.2. [They study the constants a_s = (-2)^s*A(s) = (-2)^s*L_n(B(1),...,B(s)) = (-2)^s*A330852(s)/A330860(s) for s >= 2. They also study the constants c(n) = B(n)*(-2)^n = A329001(n)/A330876(n).]
Mireille Régnier, A limiting distribution for quicksort, Informatique théorique et applications, 23(3) (1989), 335-343.
Uwe Rösler, A limit theorem for quicksort, Informatique théorique et applications, 25(1) (1991), 85-100. [He proved that M(t) has a Taylor expansion around t = 0 with an infinite radius of convergence.]
Kok Hooi Tan and Petros Hadjicostas, Density and generating functions of a limiting distribution in quicksort, Technical Report #568, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, USA, 1993; see pp. 8-11.
Kok Hooi Tan and Petros Hadjicostas, Some properties of a limiting distribution in Quicksort, Statistics and Probability Letters, 25(1) (1995), 87-94.
Vytas Zacharovas, On the exponential decay of the characteristic function of the quicksort distribution, arXiv:1605.04018 [math.CO], 2016. [The author studies the tail of phi(t) = M(i*t), where i = sqrt(-1).]

Cf. A001620, A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A329001 (numerators of B(n)*(-2)^n), A330852 (numerators of A(n)), A330860 (denominators of A(n)), A330876 (denominators of B(n)*(-2)^n), A335991 (denominators of B(n)).

For a fast Maple program for the calculation of the numbers (B(n): n >= 0), see A330852.

/* Very slow program due to recursion */
g(k) = sum(a=0, k, (-1)^a*B(k - a)/(a!*(k - a)!*2^a));
f(r) = sum(i=0, r, stirling(r + 1, i + 1, 1)*g(r - i));
b(p) = (-1)^p*(sum(r=1, p, stirling(p + 2, r + 1, 1)*B(p - r)/(p - r)!) + sum(rr=1, p-1, f(rr)*f(p - rr)) + 2*(-1)^p*p!*sum(a=1, p, (-1)^a*B(p - a)/(a!*(p - a)!*2^a)) + 2*sum(i=1, p, stirling(p + 1, i + 1, 1)*g(p - i)))/(p - 1);
B(m) = if(m==0, 1, if(m==1, 0, b(m)));
a(n) = numerator(B(n));

a(n) = numerator(B(n)), where B(n) = (n-1)!*Sum_{k=0..n-1} A(k+1)*B(n-1-k)/(k!*(n-1-k)!) for n >= 1 with B(0) = 1 and A(n) = A330852(n)/A330860(n).

Also, B(n) = c(n)/(-2)^n = A329001(n)/A330876(n)/(-2)^n.

A335991 The moment generating function of the limiting distribution of the number of comparisons in quicksort can be written in the form M(t) = m(-2t)/(exp(2gammat)Gamma(1 + 2t)) for |t| < 1/2, where m(z) = Sum_{n >= 0} B(n)z^n/n! for |z| < 1. This sequence gives the denominators of the rational numbers B(n) for n >= 0.

Original entry on oeis.org

1, 1, 4, 8, 36, 3456, 172800, 10368000, 3810240000, 177811200000, 9957427200000, 75278149632000000, 1912817782149120000000, 53023308921173606400000000, 17742659631203112173568000000000, 426249654980023566857797632000000000

Offset: 0

Petros Hadjicostas, Jul 03 2020

Despite the fact that both the numerator and denominator in the formula  M(t) = m(-2*t)/(exp(2*gamma*t)*Gamma(1 + 2*t)) each have a Taylor expansion around t = 0 with a radius of convergence equal to 1/2, the moment generating function M(t) has a Taylor expansion around t = 0 with an infinite radius of convergence. This was proved in Rösler (1991).
The formula for M(t) appears as Theorem 6.1 in Tan and Hadjicostas (1993) and derives from the work of Hennequin (1991). Hennequin conjectured a cumulant formula for the limiting distribution of the number of comparisons in quicksort in his 1989 paper, and he proved it in his 1991 thesis.
The numbers (B(n): n >= 0), with B(0) = 1 and B(0) = 0, are given (for p >= 0) by the recurrence
Sum_{r=0..p} Stirling1(p+2, r+1)*B(p-r)/(p-r)! + Sum_{r=0..p} F(r)*F(p-r) = 0, where F(r) = Sum_{i=0..r} Stirling1(r+1,i+1)*G(r-i) and G(k) = Sum_{a=0..k} (-1)^a*B(k-a)/(a!*(k-a)!*2^a).
The numbers A(n) = L_n(B(1),...,B(n)) = A330852(n)/A330860(n), where L_n(x_1,...,x_n) are the logarithmic polynomials of Bell, appear in Hennequin's cumulant formula.
Hoffman and Kuba (2019, 2020) gave an alternative proof of Hennequin's cumulant formula and gave an alternative calculation for the constants (-2)^n*A(n), which they denote by a_n. See also Finch (2020).
Hoffman and Kuba (2019-2020, Proposition 17) express the constants c(n) = B(n)*(-2)^n = A329001(n)/A330876(n) in terms of "tiered binomial coefficients". In terms of the constants c(n), the moment generating function equals M(t) = Sum_{n >= 0} (c(n)*t^n/n!)/(exp(2*gamma*t)*Gamma(1 + 2*t)) for |t| < 1/2.
Tan and Hadjicostas (1993) proved that lim_{n -> infinity} B(n)/n! = nu, where nu = 0.589164... (approximately). Also, M(-1/2) = nu*exp(gamma), where gamma = A001620 (Euler's constant).

			The first few fractions are 1/1, 0/1, 7/4, 19/8, 565/36, 229621/3456, 74250517/172800, 30532750703/10368000, 90558126238639/3810240000, ... = A335990/A335991.

Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991), p. 83.

James A. Fill and Svante Janson, Smoothness and decay properties of the limiting Quicksort density function, In: D. Gardy and A. Mokkadem (eds), Mathematics and Computer Science, Trends in Mathematics, Birkhäuser, Basel, 2000, pp. 53-64.
James A. Fill and Svante Janson, Quicksort asymptotics, Journal of Algorithms, 44(1) (2002), 4-28.
Steven Finch, Recursive PGFs for BSTs and DSTs, arXiv:2002.02809 [cs.DS], 2020; see Section 1.4. [He gives the constants a_s = (-2)^s*A(s) for s >= 2. He also calculates c(2) - c(8), where c(n) = B(n)*(-2)^n.]
P. Hennequin, Combinatorial analysis of the quicksort algorithm, Informatique théoretique et applications, 23(3) (1989), 317-333.
M. E. Hoffman and M. Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020; see Section 5.2. [They study the constants a_s = (-2)^s*A(s) = (-2)^s*L_n(B(1),...,B(s)) = (-2)^s*A330852(s)/A330860(s) for s >= 2. They also study the constants c(n) = B(n)*(-2)^n = A329001(n)/A330876(n).]
Mireille Régnier, A limiting distribution for quicksort, Informatique théorique et applications, 23(3) (1989), 335-343.
Uwe Rösler, A limit theorem for quicksort, Informatique théorique et applications, 25(1) (1991), 85-100. [He proved that M(t) has a Taylor expansion around zero with an infinite radius of convergence.]
Kok Hooi Tan and Petros Hadjicostas, Density and generating functions of a limiting distribution in quicksort, Technical Report #568, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, USA, 1993; see pp. 8-11.
Kok Hooi Tan and Petros Hadjicostas, Some properties of a limiting distribution in Quicksort, Statistics and Probability Letters, 25(1) (1995), 87-94.
Vytas Zacharovas, On the exponential decay of the characteristic function of the quicksort distribution, arXiv:1605.04018 [math.CO], 2016. [The author studies the tail of phi(t) = M(i*t), where i = sqrt(-1).]

# For a fast Maple program for the calculation of the numbers (B(n): n >= 0), see A330852.

a(n) = denominator(B(n)), where B(n) = (n-1)!*Sum_{k=0..n-1} A(k+1)*B(n-1-k)/(k!*(n-1-k)!) for n >= 1 with B(0) = 1 and A(n) = A330852(n)/A330860(n).

Also, B(n) = c(n)/(-2)^n = A329001(n)/A330876(n)/(-2)^n.

A334750 Total number of swaps needed to sort all n! permutations of n elements by the optimal dual-pivot quicksort "Count".

Original entry on oeis.org

0, 0, 4, 16, 103, 711, 5526, 48066, 463248, 4908816, 56749536, 711299232, 9609618816, 139252708224, 2154724104960, 35464952597760, 618712803717120, 11405648080834560, 221541001069731840, 4522678391979417600, 96811891510299033600, 2168416142767145779200

Offset: 0

Petros Hadjicostas, May 09 2020

nonn

The dual-pivot quicksort "Count" differs slightly from the original version of the dual-pivot quicksort algorithm of Aumüller et al. (2016, 2019). See Algorithm 1 in Neininger and Straub (2017, 2018).
By the design of the algorithm, for n >= 2, Neininger and Straub always require at least two swaps "at the end in order to bring the [two] pivots to their final positions".
Here a(n) is n! times the average number of swaps of the dual-pivot quicksort "Count" when sorting a random permutation of length n.

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.
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.

Cf. A058312, A058313, A288964, A288965, A288970, A288971.

lista(nn) = { nn = max(nn, 3); my(va = vector(nn)); va[1] = 0; va[2] = 4; for(n=3, nn, va[n] = n!*(6*sum(k=1, n-2, (n-k-1)*va[k]/k!)/(n*(n-1)) + 5*n/8 + 13/16 - 1/(16*(n - (1 + (-1)^n)/2)))); concat(0, va); };

H1(n) = sum(k=1, n, 1/k);
H2(n) = sum(k=1, n, (-1)^k/k);
H3(n) = if(0 == (n % 2), - (1/320)*(1/(n - 3) + 3/(n - 1)), (1/320)*(3/(n - 2) + 1/n));
lista(nn) = { nn = max(nn, 3); my(va = vector(nn)); va[1] = 0; va[2] = 4; va[3] = 16; for(n=4, nn, va[n] = n!*((3/4)*n*H1(n) + (1/20)*n*H2(n) - (4/5)*n + (3/4)*H1(n) + (1/20)*H2(n) - 23/160 - (-1)^n/40 + H3(n))); concat(0,va); };

Recurrence: a(n)/n! = (6/(n*(n - 1)))*(Sum_{k=0..n-2} (n - k - 1)*a(k)/k!) + 5*n/8 + 13/16 - 1/(16*(n - I[n is even])) for n >= 2 with a(0) = a(1) = 0, where I(condition) = 1 if the condition holds, and = 0 otherwise. [See p. 7 in Neininger and Straub (2017).]

a(n) = n!*((3/4)*n*Harmonic(n) + (1/20)*n*Harmonic_alt(n) - (4/5)*n + (3/4)*Harmonic(n) + (1/20)*Harmonic_alt(n) - 23/160 - (-1)^n/40 - ([n even]/320)*(1/(n - 3) + 3/(n - 1)) + ([n odd]/320)*(3/(n - 2) + 1/n)) for n >= 4, where Harmonic_alt(n) = Sum_{k=1..n} (-1)^n/n = -A058313(n)/A058312(n). [See Theorem 2.1 in Neininger and Straub (2017, 2018).]

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A334750 Total number of swaps needed to sort all n! permutations of n elements by the optimal dual-pivot quicksort "Count".

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