, when k is fixed. We give a number of results in. k q. this direction, some of which involve Eulerian polynomials and their generalizations.

Transcription

1 SOME ASYMPTOTIC RESULTS ON -BINOMIAL COEFFICIENTS RICHARD P STANLEY AND FABRIZIO ZANELLO Abstract We loo at the asymptotic behavior of the coefficients of the -binomial coefficients or Gaussian polynomials, when is fixed We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations Introduction The purpose of this note is to investigate the asymptotic behavior of the coefficients of the -binomial coefficient or Gaussian polynomial While much of the previous wor in this area has focused on the case where both a and get arbitrarily large see eg [], in this paper we will be concerned with asymptotic estimates for the coefficients of when is fixed Besides the intrinsic relevance of studying the combinatorial, analytic or algebraic properties of -binomial coefficients, our wor is also motivated by a series of recent papers that have revived the interest in analyzing the behavior of the coefficients of, as well as their applications to other mathematical areas See for instance [7], where I Pa and G Panova have first shown algebraically the strict unimodality of, as well as the subseuent combinatorial proofs of the Pa-Panova result by the second author of this paper [3] and by V Dhand [3] See also another interesting recent wor by Pa and Panova [8] as well as their extensive bibliography, where the coefficients of have been investigated in relation to uestions of representation theory concerning the growth of Kronecer coefficients Further, one of the results of this note, Theorem 22, has also been motivated by, and finds a first useful application in the study of the unimodality of partitions with distinct parts that are contained inside certain Ferrers diagrams see our own paper [0] For m = a/2 the middle exponent of when or a are even, and the smaller of the two middle exponents otherwise, define g,c a to be the coefficient of degree m c of, and let f,ca = g,c a g,c+ a Our first main result is a description of the generating functions in two variables, referring to a and c of g,c a and f,c a In 200 Mathematics Subject Classification Primary: 05A6; Secondary: 05A7 Key words and phrases -binomial coefficient; asymptotic enumeration; integer partition; Eulerian number; Euler-Frobenius number; Kosta number This author s contribution is based upon wor supported by the National Science Foundation under Grant No DMS This author is partially supported by a Simons Foundation grant #274577

2 2 RICHARD P STANLEY AND FABRIZIO ZANELLO particular, it follows from our result that both g,c a and f,c a are uasipolynomials in a, for any given and c Our next result, Theorem 24, is an asymptotic estimate of the coefficient of degree αa c of, when a, for any given integer c, positive integer, and nonnegative real number α Quite surprisingly, this result connects in a nice fashion to Eulerian numbers and, more generally, to Euler-Frobenius numbers, as we will discuss extensively after the proof of the theorem Finally, our last main result, Theorem 26, presents an asymptotic estimate of the difference between consecutive coefficients of, again for fixed We will wrap up this note with a brief remar, in order to highlight an interesting connection of our last result with Kosta numbers and to present some suggestions for further research 2 Some asymptotic properties of the coefficients of In this section, we study the asymptotic behavior of the coefficients of for fixed Given, c 0, and a 0, set m = a/2 Define g,c a = [ m c ], f,c a = g,c a g,c+ a, where [ n ]F denotes the coefficient of n in the polynomial or power series F Lemma 2 Let F C[[]], and c,j,i Z with j > i 0 We have: a b a 0 a 0 c 0 [ aj c ] ai Fx a = j i ζ j i = [ aj c ] ai Fx a t c = j i ζx c Fζx ζ j i = Proof For any G = a i i C[[]] and h, write D h G = a hi x hi, Fζx ζxt x x /j i x x /j i the hth dissection of G It is an elementary and standard result see eg [9, Exercise 60] that D h G = Gζx h The sum is over all h complex numbers ζ satisfying ζ h = Hence a follows ζ h =

3 SOME ASYMPTOTIC RESULTS ON -BINOMIAL COEFFICIENTS 3 Part b is the generating function in t with respect to c of the formula of part a We have: a 0 c 0[ aj c ] ai Fx a t c = [ aj i ] c Fx a t c a 0 c 0 and the proof follows = a 0 = j i [ aj i ] F t xa ζ j i = Fζx ζxt x x /j i From Lemma 2, it is easy to describe the form of the generating functions for g,c a and f,c a, when and c are fixed For this purpose, define a uasipolynomial to be a function h: N C where N = {0,,2} of the form hn = c d nn d +c d nn d + +c 0 n, where each c i n is a periodic function of n If c d n 0 then we call d the degree of h For more information on uasipolynomials, see for instance [9, 44] Write F x,t = f,c ax a t c a 0 c 0 G x,t = g,c ax a t c a 0 c 0 Theorem 22 Fix and set j = /2 If we denote both F and G by H, then N x,t D x tx t 2 x t 3 x t j x, even H x,t = N x,t D x tx 2 t 3 x 2 t 5 x 2 t x 2, odd where N x,t Z[x,t] and D x is a product of cyclotomic polynomials In particular, for fixed and c we have that g,c a and f,c a are uasipolynomials Proof Case : = 2j We have m = a/2 = aj Write 2 a+ a+2 = i P i ai, where P i is a polynomial in independent of a Specifically, we have 3 P i = S [] #S=i s S s,

4 4 RICHARD P STANLEY AND FABRIZIO ZANELLO Writing []! = 2, we get G x,t = [ m c ] x a t c a 0 c 0 = [ aj c ] i P i ai x a t c []! a 0 c 0 = [ aj c ] []! i P i ai x a t c a 0 c 0 The proof now follows from Lemma 2a Note in particular that the expression Fζx in Lemma 2 will produce cyclotomic polynomials in the denominator of F x,t, while the denominator ζxtinpart b will lead to thefactor t j i xin the denominator of F x,t The proof for F x,t is completely analogous Case 2: = 2j+ The proof is analogous to Case Now we have to loo at a = 2b and a = 2b+ separately When a = 2b we get that the part of G x,t with even exponent of x is G x,t = a 0 c 0 [b c ] 2b+ x 2b t c When we apply Lemma 2, the denominator term becomes ζx 2 t, where ζ j i = and j i is odd This produces a factor t j i x 2 where j i is odd in the denominator of F x,t Exactly the same reasoning applies to a = 2b+, so the proof follows Example 23 Write Φ m x for the mth cyclotomic polynomial normalized to have constant term Hence Φ x = x, Φ 2 x = +x, Φ 3 x = +x+x 2, etc One can compute the following: 4 F 3 x,t = G 3 x,t = F 4 x,t = G 4 x,t = +tx+tx 3 +t 3 x 4 x+x+x 2 tx 2 t 3 x 2 N 3 x,t x 2 x 4 tx 2 t 3 x 2 tx+t 2 x 2 x 2 x 3 tx t 2 x x++tx 2 t+t 2 x 3 x 2 x 2 x 3 tx t 2 x F 5 x,0 = x5 x 6 +x 7 +x 2 Φ 3 Φ3 2 Φ 3Φ 2 4 Φ 6Φ 8 B 5 x G 5 x,0 = x 2 x 4 x 6 x 8 M 6 x,t F 6 x,t = Φ 4 Φ 2 2Φ 3 Φ 4 Φ 5 tx t 2 x t 3 x N 6 x,t G 6 x,t = Φ 6 Φ3 2 Φ 3Φ 4 Φ 5 tx t 2 x t 3 x,

5 SOME ASYMPTOTIC RESULTS ON -BINOMIAL COEFFICIENTS 5 where N 3 x,t = tx+ t+t 2 x 2 +t t 2 x 3 t t 2 x 4 t 2 +t 3 x 5 B 5 x = x+2x 2 +x 3 +2x 4 +3x 5 +x 6 +5x 7 +x 8 +3x 9 +2x 0 +x +2x 2 x 3 +x 4 +3x 0 +x 2 x 3 +2x 4 x 5 +x 7 2x 8 +x 9 M 6 x,t = + t t 2 x t t 3 t 4 x 2 t t 2 t 3 t 4 +t 5 x 3 2t t 2 +t 5 x 4 2t t 2 +t 3 +t 4 x 5 +t+t 2 t 3 2t 4 +t 5 x 6 + t 3 2t 4 +t 5 x 7 + t t 2 t 3 t 4 +t 5 x 8 t+t 2 t 4 x 9 +t 3 +t 4 t 5 x 0 t 5 x N 6 x,t = ++2t+t 2 x t t 2 2t 3 t 4 x t 2 3t 3 t 4 +t 5 x t 2 4t 3 t 4 +2t 5 x 5 +4 t 4t 2 4t 3 t 4 +3t 5 x 6 +3 t 5t 2 4t 3 +3t 5 x 7 + t 4t 2 3t 3 +t 4 +4t 5 x 8 2t 2 +t 3 t 4 3t 5 x 9 + t 2 t 3 +3t 5 x 0 The denominator of F 8 x,t is given by and that of G 8 x,t by t+t 2 +t 3 t 4 2t 5 x +t 3 +t 4 +t 5 x 2 Φ 6 Φ3 2 Φ2 3 Φ 4Φ 5 Φ 7 tx t 2 x t 3 x t 4 x, Φ 8 Φ3 2 Φ2 3 Φ 4Φ 5 Φ 7 tx t 2 x t 3 x t 4 x Let us also note that F 8 x,0 = a+8 a 0[ 4a ] [ 4a ] 8 x a = +x x3 x 4 +x 6 +x 7 +x 8 +x 9 +x 0 x 2 x 3 +x 5 +x 6 +x x 2 x 3 2 x 4 x 5 x 7 = +x 2 +x 3 +2x 4 +2x 5 +4x 6 +4x 7 +7x 8 +8x 9 +2x 0 + This generating function appears in a paper [5, p 847] of Igusa, stated in terms of the representation theory of SLn,C Igusa also computes F 2 x,0, F 4 x,0, and F 6 x,0 From the techniues for computing F x,t and G x,t, we can determine asymptotic properties of some of the coefficients of, for fixed The coefficients of have been considered for a, by Taács [] and others, but the computation for fixed seems to be new

6 6 RICHARD P STANLEY AND FABRIZIO ZANELLO Theorem 24 Fix α 0 α R, c Z, and a positive integer Then [ αa c ] =!! Cα,a +Oa 2, where α Cα, = i α i i Proof First assume that α is rational, say α = u/v Fix 0 r < v and consider only those a of the form a = vb+r Set d = ur/v Thus [ 5 ] ua/v c = [ ub+d c ] vb+r+ vb+r+ vb+r+ Write G α,,r x = a 0 a rmodv [ ] ua/v c x a = vb+r+ b 0[ ub+d c ] x vb+r We now apply euation 5, expand the numerator and apply Lemma 2a We obtain a linear combination of expressions lie ζx e 6 s ζx ζ 2 x 2 ζ x = Gx x x /s, ζ s = x x /s say Let ζ s = e 2πi/s, a primitive sth root of unity The order to which is a pole in euation 6 is thus at most the order to which ζ s is a pole of Gx Now any term indexed by ζ has ζ s as a pole of Gx of order less than, while the term indexed by ζ = has a pole of order at most at x = Hence if in the end we have a pole of order, then it suffices to retain only the term in 6 indexed by ζ = Therefore if, for any integer e, ζx e c 0 s ζx ζ 2 x 2 ζ x = ζ s = x x /s x +O, x then Write c 0 = lim x s x s x x 2 x = s! x e vb+r+ vb+r+ vb+r+ = i Q i bvi, []!

7 SOME ASYMPTOTIC RESULTS ON -BINOMIAL COEFFICIENTS 7 where Q i is a polynomial independent of b and v, so Q i = i Note that u vi 0 if and only if i α It follows that G α,,r x = [ ] ub+d c i Q i bvi []! b 0 = b 0[ u vib ] i Q i c d []! = α i u vi! i = α i u vi! i x bv+r x bv+r xr x v +O v x +O x x Now sum over 0 r < v Since we have v terms in the sum, we pic up an extra factor of v on the right, giving [ ua/v ] x a = α i u vi a 0! i v x +O x = α i α i! i x +O x Now [x a ] = x +a = a! +Oa 2, completing the proof for α rational The proof for general α now follows by a simple continuity argument, using the unimodality and symmetry of the coefficients of The numbers Cα, have appeared before and are nown as Euler-Frobenius numbers, denoted A, α,α α For a discussion of the history and properties of these numbers, see Janson [6] Some special cases are of interest Recall that the Eulerian number Ad, i can be defined as the number of permutations w of,2,,d with i descents eg [9, 4] Similarly the MacMahon number Bd, i can be defined as the number of elements in the hyperoctahedral group B n according to the number of type B descents For further information, see [] Standard results about these numbers imply that for integers j <, Cj, = A,j, 2 C2j /2, = B,j

8 8 RICHARD P STANLEY AND FABRIZIO ZANELLO There is an alternative way to show the above formula for Cα, done with assistance fromfuliu Write β = α Since thecoefficient of aβ in isthe number of partitions of aβ into at most a parts of length at most, euivalently, it is eual to the number of solutions m,,m in nonnegative integers to m +2m 2 + +m = aβ, m + +m a Set x i = m i /a and let a Standard arguments see eg, [9, Proposition 463] show that Cα, is the -dimensional relative volume as defined in [9, p 497] of the convex polytope: x +2x 2 + +x = β, x +x 2 + +x x i 0, i, Set y i = x i +x i+ + +x The matrix of this linear transformation has determinant, so it preserves the relative volume We get the new polytope P defined by y +y 2 + +y = β, 0 y y 2 y By symmetry, the relative volume of P is /! times the relative volume of the polytope y +y 2 + +y = β, 0 y i, i This polytope is a cube cross-section, whose relative volume is computed eg in [6, Theorem 2], completing the proof When α Q, Cα, is related to the Eulerian polynomial A x via the following result Proposition 25 Let v P Then 7 v u 0Cu/v,x u = +x+x 2 + +x v A x Proof We have 8 v u 0 Cu/v,x u = u 0 u/v i u vi x i u A fundamental property of Eulerian polynomials is the identity see [9, Proposition 44] n x n = A x x n 0

9 SOME ASYMPTOTIC RESULTS ON -BINOMIAL COEFFICIENTS 9 Hence, 9 A x+x+ +x v = x v n 0n x n It is now routine to compute the coefficient of x m on the right-hand sides of euations 8 and 9 and see that they agree term by term Note that if j P and we tae the coefficient of x jv on both sides of euation 7, then we obtain the identity v A,j = [x vj ]+x+x 2 + +x v A x It is not difficult to give a direct proof of this identity Let us now turn to the difference between two consecutive coefficients of, ie, the function f,c a of euation We consider here only the coefficients near the middle ie, aj when = 2j, though undoubtedly our results can be extended to other coefficients Note that, by the previous theorem, we have [ aj c ] [ aj c ]!! Cα,a, a Thus we might expect that the difference [ aj c ] [ aj c ] grows lie a 2 However, the next result shows that the correct growth rate is a 3 Theorem 26 Let c N and = 2j, where j P Then for j 3 we have [ aj c ] [ aj c ] = 2c+ 3!! Da 3 +Oa 4, where Proof Write D = j i+ j i 3 2 i F x,t = [ aj c ] [ aj c ] x a t c a 0 c 0 = j 0[ aj a+ ] 2 t xa t c When 6, the order to which a primitive sth root of unity x = ζ s is a pole of F x,t is at most 3 Thus we need to show that the pole at x = contributes the stated result Let F x,t = α t x +β t x +O 2 x 3

10 0 RICHARD P STANLEY AND FABRIZIO ZANELLO First we show that α t = 0 Reasoning as in the proof of Theorem 24 gives α t = j 2!! t i i j i 2 Since is even, the summand i i j i 2 remains the same when we substitute i for i Moreover, when i = j the summand is 0 Hence α t = 2 2!! t i j i 2 i This sum is the th difference at 0 of a polynomial of degree 2, and is therefore eual to 0 see [9, Proposition 92], as desired We now need to find the coefficient β of x 2 in the Laurent expansion at x = of linear combinations of rational functions of the type H = Px x 2 x xt = α x + β +, x 2 where Px is a polynomial in x Write i x = +x+x 2 + +x i It is easy to see that α = P/! t Thus β = lim x 2 Px x x 2 x xt P! t x = lim x x Px! t P2 x x xt 2 x x xt! t d =! 2 t 2 dx Px! t P2 x x xt x= = P! t!p 2 t+pt! 2 t 2 =! t 2 =! t 2 P t 2 P t+pt P t+ 4 P +t+2 2 t Pt Let us apply this result to Px = P i x, where P i is defined by euation 3 Clearly P i = i, while P i = s S [] s S #S=i The element i [] appears in i i-element subsets of [] Hence + i = i = i 2 P i= i,

11 SOME ASYMPTOTIC RESULTS ON -BINOMIAL COEFFICIENTS where when i = 0 we set = 0 Arguing as in the proof of Theorem 24 now gives j β t = i+ j i 3! t 2 i 2 t j i +t t t 2 i If we set t = on the right-hand-side of euation 0, then a straightforward computation shows that the sum is 0 If we set t =, then another computation gives j j i 3 i+ i Since the proof now follows +t t 2 = c 0 2c+t c, Remar 27 a It follows from wor of Verma [2] and of Hering and Howard [4] that D also satisfies K a/2,/2,a = 3! Da 3 +Oa 4, where K λµ is a Kosta number and a denotes the partition of a with a s Is the appearance of D in both Theorem 26 and euation just a coincidence? b Theorem 26 is false for j = 2 Indeed, it follows from euation 4 that t+t 2 F 4 x,t = 6 t t 2 x +O 2 x and [ 2a c ] [ 2a c ] a+4 4 = 24 2c++3 c a+o, a c An obvious problem arising from our wor is the extension of Theorem 24 to additional terms Can such a computation be automated? 3 Acnowledgements We are grateful to several anonymous reviewers for several comments, to Fu Liu for her assistance with the proof presented after Theorem 24, and to Qinghu Hou for pointing out some errors in our original computations and for noting that the Omega Pacage [2] of Andrews, Paule, and Riese can be used very effectively for the computation of F x,t and G x,t The second author warmly thans the first author for his hospitality during calendar year 203 and the MIT Math Department for partial financial support

12 2 RICHARD P STANLEY AND FABRIZIO ZANELLO References [] A06087, On-Line Encyclopedia of Integer Seuences Available at [2] G E Andrews, P Paule and A Riese: MacMahon s partition analysis: the Omega Pacage, European J Combin , [3] V Dhand: A combinatorial proof of strict unimodality for -binomial coefficients, Discrete Math , [4] M Hering and B Howard: The ring of evenly weighted points on the line, Math Z , no 3 4, [5] J-I Igusa: Modular forms and projective invariants, Amer J Math , [6] S Janson: Euler-Frobenius numbers and rounding, Online J Analytic Comb 8 203, 34 pp [7] I Pa and G Panova: Strict unimodality of -binomial coefficients, C R Math Acad Sci Paris , no 2, [8] I Pa and G Panova: Bounds on the Kronecer coefficients, preprint Available on the arxiv [9] R Stanley: Enumerative Combinatorics, Vol I, Second Ed, Cambridge University Press, Cambridge, UK 202 [0] R Stanley and F Zanello: Unimodality of partitions with distinct parts inside Ferrers shapes, European J Combin , [] L Taács: Some asymptotic formulas for lattice paths, J Stat Planning and Inference 4 986, [2] D-N Verma: Toward classifying finite point-set configurations, preprint 997 [3] F Zanello: Zeilberger s KOH theorem and the strict unimodality of -binomial coefficients, Proc Amer Math Soc , no 7, Department of Mathematics, MIT, Cambridge, MA address: rstan@mathmitedu Department of Mathematical Sciences, Michigan Tech, Houghton, MI address: zanello@mtuedu