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[Nov. EULERIAN NUMBERS AND THE UNIT CUBE DOUGLAS HENSLEY Texas A & M University, (Submitted 1. College Station, April 1981) TX 77843 INTRODUCTION There is an excellent expository paper [3] on Eulerian numbers and polynomials, and we begin with a quotation from it: "Following Euler [5] we may put -Lz_L« E nn^ where Hn = Hn(X) (x * i), (i.D is a rational function of A; indeed Rn = 2?n(X) - (X - DnHn(X) (1.2) is a polynomial in X of degree n - 1 with integral coefficients. Rn = E^JfcX*" 1 (" > 1). If we put d.3) k = l then the first few values of An^ denotes the row and k the column; 1 1 1 1 1 1 4 11 26 57 are given by the following table, where n 1 11 66 302 1 26 302 (1.4) 1 57 Alternatively, Worpitzky showed that the An\ 1 may be defined by means of The numbers Ank occur in connection with Bernoulli numbers and polynomials [11], and splines [10], and as the number of permutations of (1, 2, .. . , ri) with k vises. [A permutation (<z19 ..., ccn) has a rise at ai if a^ < ai + 1; by convention, there is a rise to the left of a1.] The ^4nfe satisfy a recursion and are symmetric: K + i,k = ^n,k + (n -fe+ i M ^ ^ i 344 (1.6) 1982] EULERIAN NUMBERS AND THE UNIT CUBE 345 and A n,k = An,n-k (1 < fe < n ) . +l From (1.6), it follows that n £ Ak = n\ (n > 1). We now consider the unit cube Qn : 0 < 2^ < 1 (1 < £ < n) , with the usual measure. It is evident from elementary calculations and from observation of (1.4) that, for n = 2, 3, or 4 and 1 < k < n, the volume F„^ of the section n fc - 1 < Yixi < fc i=l of the unit cube is given by 7n& = Any. /n\ . This observation led Hillman (in a private communication with this author) to conjecture that, generally, V nk = A nk/nl He was right. 2. APPLICATIONS In the notation of Section 1, we have THEOREM 1: For 1 < k < n, p/e £ave 7nfc = Ank/nl (2.1) The proof is not difficult, but we defer that to the last. What is nice about this is that the unit cube is the natural probability space for a sum of n independent random variables X^ (1 < i < n) identically and uniformly distributed on [0, 1]. Thus, we may reinterpret (2.1) to read: For 1 < k < n, Probffe - 1 < £ Xt < k J = Ank /nl (2.2) Through this interpretation, the central limit theorem and related results can be brought to bear on the asymptotic behavior of the Eulerian numbers. For instance, the variance of each X^ is •l 2 / ' (x - l/2) dx = 1/12. '0 71 Thus the variance of £ Xi x is fixed and Jo is nI'12. Now, by the central limit theorem, if x (n/12)1/2x + then \n, EULERIAN NUMBERS AND THE UNIT CUBE 346 [Nov. f e~t2/2dt. lim ProbjE X. < oon ) = — (2.3) n Since the probability density function / (£) of 2 X± tends to zero uniformly in t as n ->• °° 9 we can replace 0)n with [u)w] 'in (2.3). have Then, from = - ^ / e't/2dt. lim J^A^/nl (2.2), we (2.4) This is equivalent to Theorem 1 of [4], It may be that this approach permits a simpler proof or an improvement in the error term in the other theorem of [4]9 which states that (l/nl)AnA.n] = (6/n^)1/2exp(-|*2) +0(n" 3/!+ ). (2.5) From a geometric point of view, one important property of the cube is that it is convex. The Brunn-^Minkowski theorem' states fzh^t,|\the" area A(t) of the intersection of a hyperplane H(t) with equation n 1 with a convex body Q in real n-space has a concave nth root on the interval where it is positive. Thus, if Hn(t) has equation n 1 and An{t) is the area of Hn(t) C] Qn (where Qn is still the unit cube 0 < ^ < 1, 1 < i < n) , then {An(t))1/n is concave on (0, n) . Consequently, log An(t) is concave on (0, n). (2.6) There is a simple relation between i4n(t)andthe probability density function fn(t) ofS^: i4n(t) = ,/nfn(t). (See, e.g., [6].) Now let Fn^ be the volume of Qn between H(k - 1) and H(k). Vnk = n~1/2 fk A -1 Then, fk An(t)dt =/ fn(t)dt. (2.7) A -1 There is a considerable literature on logarithmic concavity. A function g(t) is called tog-concave if #(£) ) 0 on R and is positive on just one interval, and if log git) is concave on that interval. A very special case of a theorem due to Prekopa says that if fit) is log-concave, then 1982] EULERIAN NUMBERS AND THE UNIT CUBE Fix) = f 347 f(t)dt JX - O is also log-concave [2, 8, 9]. In particular, V(x) = n-ll2f A(t)dt •* X - Q is log-concave, and in most particular, \ , *.!*;, *+i < vlk> (2-8) or what is the same thing, Antk.1Antk +1 <A2n,k. (2.9) This is due to Kurtz, who proved strict inequality in (2.9) when 1 < k ^ n. 3. PROOF OF THEOREM 1 n The probability density functions fn (t) for £ X^ can be generated recur1 sively starting with J l V W (1 \o if 0 < t < 1 otherwise and using f„ + 1(*> = /„(*) * A C * ) - / /„(w)A<* - " ) d M = / Jo /„("><*"• (3-1) Jt-i Thus, ^ = / fn(t)dt = /n+1(fc). (3.2) It follows from (1.5) (but not trivially) that k-i &nk = E (-DJ'(n *il)fr- J) n . (3.3) j-o This is (2.15) of showing that [3] and is due to Euler. fn + iW Now, fn+1(t) transform is = -V ^(-W^ Thus, we can prove Theorem 1 by * X)0c - J)n. is the convolution of n + 1 copies of f±(t)s (3.4) so its Laplace (I \n+1 F( 7 (s) = (^(1 - £"*)) . (See, e.g., [1].) Expanding (3.5) by the binomial theorem gives (3.5) 348 EULERIAN NUMBERS AND THE UNIT CUBE j-o x J [Nov. ' and the inverse Laplace transform of the sum of these n + 2 terms computes to fn+1(t) = E ^ ( - i ) j n - > - ^ : . where (t - J ) + is 0 for t < j and t - j for t > j.- With t = k, to (3.4). • (3.6) (3.6) reduces REFERENCES 1. W. Boyce & R* DiPrima. Elementary Differential Equations and Boundary Value Problems* Chap. 6. New York: John Wiley & Sons, 1977. 2. H. Brascamp & E. Lieb. "On Extensions of the Brunn-Minkowski and Prekopa-Leindler Theorems, Including Inequalities for Log Concave Functions, and With an Application to the Diffusion Equation." J. Func. Anal. 22 (1976):366~389. 3. L. Carlitz. "Eulerian Numbers and Polynomials." Math. Mag. 33 (1959): 247-260. 4. L. Carlitz, D. Kurtz, R. Scoville,& 0. Stackelberg. "Asymptotic Properties of Eulerian Numbers." Z. Wahrscheinlichkeitstheorie verw. Geb. 23 (1972):47-54. 5. Le Euler. "Institutiones Calculi Differentialis." Omnia Opera (1), 10 (1913). 6. D. Hensley. "Slicing the Cube in R" and Probability (Bounds for the Measure of a Central Cube Slice in R n by Probability Methods)." Proc. Am. Math. Soc. 73, No., 1 (1979): 95-100. 7. A. Hillman, P. Mana, & C. McAbee. "A Symmetric Substitute for Stirling Numbers." The Fibonacci Quarterly 9, no. 1 (1979):51-60, 73. 8. A. Prekopa. "On Logarithmic Concave Measures and Functions." Acta Sci. Math. (Szeged) 34 (1973):335-343. 9. Y. Rinott. "On Convexity of Measures." The Annals of Probability 4, No. 6 (1976):1020-1026. 10. I. Schoenberg. "Cardinal Spline Interpolation and the Exponential Euler Splines." Lecture Notes in Math., No. 399 (New York: Springer, 1973): 477-489. 11. H. Vandiver. "An Arithmetical Theory of the Bernoulli Numbers." Trans. Am. Math. Soc. 51:502-531. A A "fit "k A"