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A329001
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Numerator of the rational constant term c(n) of the n-th moment mu(n) of the limiting distribution of the number of comparisons in quicksort (for n >= 0).
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7
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1, 0, 7, -19, 2260, -229621, 74250517, -30532750703, 90558126238639, -37973078754146051, 21284764359226368337, -1770024989560214080011109, 539780360793818428471498394131, -194520883210026428577888559667954807, 911287963487139630688627952818633149408727
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OFFSET
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0,3
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COMMENTS
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Let X_n be the number of comparisons needed to sort a list of n distinct numbers by quicksort. Then E(X_n) = A115107(n)/A096620(n) = -4*n + + 2*(1+n)*HarmonicNumber(n) or E(X_n) = A093418(n)/A096620(n) = -3*n + + 2*(1+n)*HarmonicNumber(n) depending on the assumptions.
Régnier (1989) and Rösler (1991) proved that (X_n - E(X_n))/n converges a.s. (and in other modes of convergence) to a non-degenerate r.v. X. Rösler proved the existence of all moments for X and Tan and Hadjicostas (1995) proved that it has a density w.r.t. to the Lebesgue measure. Fill and Janson (2000, 2002) proved many other properties of the distribution of X.
Hennequin (1989, 1991) calculated moments and cumulants of X. He proved that mu(n) = E(X^n) is a linear combination of a finite number of zeta values at positive integer arguments with rational coefficients plus a rational constant c(n). It is the numerators of this constant term c(n) of mu(n) that we tabulate here, while the denominators are in A330876.
Hennequin (1991) proved that the n-th cumulant of X for n >= 2 is k(n) = (-2)^n*(A(n) - (n-1)!*zeta(n)), where A(n) = A330852(n)/A330860(n) with A(0) = 1 and A(1) = 0. Also, (-2)^n*A(n) = A330875(n)/A330876(n); see Hoffman and Kuba (2019-2020) and Finch (2020).
Actually, Hoffman and Kuba (2019-2020, Proposition 17) express the constants c(n) in terms of "tiered binomial coefficients". Finch (2020) uses their results to write a Mathematica program with which he calculates at least c(2) - c(8). The values c(2) - c(8) have also been calculated indirectly by Ekhad and Zeilberger (2019).
Because moments and cumulants are connected via the relationship mu(n) = Sum_{s=0..n-1} binomial(n-1,s)*k(s+1)*mu(n-1-s) (e.g., see Tan and Hadjicostas (1993)), we easily deduce that c(n) = Sum_{s=0..n-1} binomial(n-1,s)*(-2)^(s+1)*A(s+1)*c(n-1-s) for n >= 1 because c(1) = A(1) = mu(1) = k(1) = 0 and k(n) = (-2)^n*A(n) - (-2)^n*(n-1)!*zeta(n) for n >= 2 and c(n) is the constant term of mu(n).
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REFERENCES
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Pascal Hennequin, Analyse en moyenne d'algorithmes, tri rapide et arbres de recherche, Ph.D. Thesis, L'École Polytechnique Palaiseau (1991).
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LINKS
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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 the first eight asymptotic moments mu(n) can be derived.]
Steven Finch, Recursive PGFs for BSTs and DSTs, arXiv:2002.02809 [cs.DS], 2020; see Section 1.4. [He gives a Mathematica program to generate the constants c(n).]
Uwe Rösler, A limit theorem for quicksort, Informatique théorique et applications, 25(1) (1991), 85-100. [In Section 5, he gives a recursion for the moments mu(n), which potentially can be used to find the constants c(n), but he does not do any explicit calculations.]
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FORMULA
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Recurrence for the fractions: c(n) = Sum_{s=0..n-1} binomial(n-1,s)*(-2)^(s+1)*A(s+1)*c(n-1-s) for n >= 1, with c(0) = 1 = A(0) and c(1) = 0 = A(1), where A(n) = A330852(n)/A330860(n) and (-2)^n*A(n) = A330875(n)/A330876(n).
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EXAMPLE
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The first few c(n) are {1, 0, 7, -19, 2260/9, -229621/108, 74250517/2700, -30532750703/81000, 90558126238639/14883750, -37973078754146051/347287500, ...}.
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MAPLE
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# The function A is defined in A330852.
# Produces the constants c(n):
c := proc(p) local v, a: option remember:
if p = 0 then v := 1; end if: if p = 1 then v := 0: end if:
if 2 <= p then v := (p - 1)!*add((-2)^(a + 1)*A(a + 1)*c(p - 1 - a)/(a!*(p - 1 - a)!), a = 0 .. p - 1): end if:
v: end proc:
# Produces the numerators of c(n):
seq(numer(c(n)), n=0..15);
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MATHEMATICA
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(* The function A is defined in A330852. *)
c[p_] := c[p] = Module[{v, a}, If[p == 0, v = 1; ]; If[p == 1, v = 0]; If[2 <= p, v = (p - 1)!*Sum[(-2)^(a + 1)*A[a + 1]*c[p - 1 - a]/(a!*(p - 1 - a)!), {a, 0, p - 1}]]; v];
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CROSSREFS
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Cf. A063090, A067699, A093418, A096620, A115107, A288964, A288965, A288970, A288971, A330852, A330860, A330875, A330876 (denominators).
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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