Congruences modulo high powers of 2 for Sloane s box stacking function

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1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume (009), Pages 55 6 Congruences modulo high powers of for Sloane s box stacking function Øystein J. Rødseth Department of Mathematics University of Bergen, Johs. Brunsgt., N 5008 Bergen Norway rodseth@math.uib.no James A. Sellers Department of Mathematics Penn State University, University Park, PA 680 U.S.A. sellersj@math.psu.edu Dedicated in honor of George Andrews 70 th birthday Abstract We are given n boxes, labeled,,...,n.boxiweighs i grams and can support a total weight of i grams. The number of different ways to build a single stack of boxes in which no box will be squashed by the weight of the boxes above it is denoted by f(n). Ina006paper,thefirstauthor asked for congruences for f(n) modulo high powers of. In this note, we accomplish this task by proving that, for r 5andalln 0, f( r n) f( r n) 0 (mod r ), and that this result is best possible. Some additional complementary congruence results are also given. Introduction In 005, Sloane and Sellers [8] considered the following problem: Suppose we have boxes with labels,,,...n. A box labeled i weighs i grams and can support a total weight of i grams. We wish to build single

2 56 ØYSTEIN J. RØDSETH AND JAMES A. SELLERS stacks of boxes with distinct labels in such a way that no box will be squashed by the weight of the boxes above it. What is the number f(n) of different ways to build such a single stack of boxes? This function f(n) is the function identified in the title as Sloane s Box Stacking Function and is the primary object of interest in this work. By way of example, note that f() = where the admissible stacks are the following: The other two possible stacks are excluded since + > and the box labeled would collapse in both cases. The first few values of f(n), for n =0,,,..., are,,, 8,,, 6, 5, 78, 09, 9, 99, 6, 9,, 58, 686. Additional values of f(n) are given by Sloane [7, Sequence A08905]. Sloane and Sellers [8] extensively studied f(n). One of their main results is that the generating function F (q) = n=0 f(n)qn can be written as ( ) F (q) = ( q) q q + i= q i i j=0 ( qj ). () In a more recent work, Rødseth [] gave a simple proof of (). At the end of his note, he asked for congruences for f(n) modulo (high) powers of. In this work we satisfy that request by proving the following theorem. Theorem For r 5 and all n 0, f( r n) f( r n) 0 (mod r ). Moreover, no higher power of divides f( r n) f( r n) if n is odd. Some complementary results are given at the end of Section 5; for example, we show that, for all n 0, f( n) f( n) 0 (mod 5 ). We note that Theorem is similar to families of congruences satisfied by related functions. For example, Churchhouse [] conjectured, and Rødseth [] proved, that

3 CONGRUENCES MODULO HIGH POWERS OF 57 the binary partition function (the function which counts the partitions of an integer n using only powers of as parts) satisfies a similar set of congruences; also see [, Section 0.]. More recently, a similar family of results was proven for the function b(n), originally defined by Sloane and Sellers [8], which counts the number of non-squashing partitions of n into distinct parts. See [6] for more details on these congruences satisfied by b(n). Indeed, we will prove Theorem using tools developed in [5, 6]. However, for r<5, the difference f( r n) f( r n) behaves rather irregularly, and this creates some extra difficulties which we overcome below. These irregularities make the search for a result like Theorem more difficult, especially as compared to the results in [, ] and [5, 6]. The structure of this paper is very straightforward. In Section we introduce some auxiliaries which are necessary for the proof of Theorem. Theorem is then proven in Section with most of the technical details postponed for the sake of readability. Those necessary details are then given in Sections and 5. Auxiliaries The power series in this paper will be elements of Z[[q]], the ring of formal power series in q with coefficients in Z. We define a Z-linear operator U, acting on Z[[q]], by U a(n)q n = a(n)q n. n n Notice that if A(q),C(q) Z[[q]], then U(A(q)C(q )) = (UA(q))C(q). () In the following, a power series A(q) Z[[q]] will often be written as A. If the argument is not q, then we will, of course, include the argument in the notation. We shall use below the following identity for binomial coefficients: ( ) r/ n + r ( )( ) r i n + r i = ( ) i r i. () r i r i The truth of this relation follows by expanding both sides of the identity ( + q) n =(+q( + q)) n and comparing the coefficient of q r on each side of the equation. Let q h i =, i 0. ( q) i+ Then ( ) n + i h i = q n, () i n=

4 58 ØYSTEIN J. RØDSETH AND JAMES A. SELLERS so that Uh r = It follows from () and () that ( ) n + r q n. r n= r/ ( ) r i Uh r = ( ) i r i h r i (5) i for r 0. Simple calculations show that Uh 0 = h 0, (6) Uh = h, (7) Uh = h h. (8) Also notice that U q = q. (9) We recursively define functions L r and M r for r by ( ) L = h and L r+ = U q L r (0) and M =h and M r+ = L r+ (UL r ) q + UM r. () The motivation for these definitions will become clear in the following section. Proof of Theorem In this proof we will use the auxillary function B(q) = q + q i i j=0 ( qj ). () i= This function is the generating function of b(n), mentioned in the Introduction. It was introduced and studied by Sloane and Sellers [8] in connection with their study of f(n). In particular, they showed that B(q) satisfies the functional equation B (q) = q B (q )+ q q, () where B = B. By () and (), we have F (q) = ( q) B (q)+ q,

5 CONGRUENCES MODULO HIGH POWERS OF 59 which we can write as F = ( h + ) B (q)+ q q. () Using (), we get ( F = h + h + ) B (q )+h q q + q. Next, we apply the operator U. Using the linearity of U along with () and (7) (9), we obtain ( UF = h + h + ) B +h + q q, so that, by (), UF F =h B +h. (5) Recalling the definitions of L r and M r given in (0) and (), we now prove by induction on r that, for all r, U r F U r F = L r B + M r. (6) By (5), we know that (6) is true for r =. Suppose that (6) holds for some r. By (), we then have ( U r F U r F = L r q B (q )+ q ) + M q r ( ) = q L r B (q )+ q L r L r q + M r, and applying U and using (), we get by (0) and (), U r F U r F = L r+ B + L r+ (UL r ) q + UM r = L r+ B + M r+. Thus (6) holds for all r. For r 6, we have, by Lemma in Section, and by Lemma in Section, By (6), we thus have L r 0 (mod r+ ), M r r h (mod r ). U r F U r F r h (mod r ), and the proof of Theorem is easily completed.

6 60 ØYSTEIN J. RØDSETH AND JAMES A. SELLERS Three Technical Lemmas We have chosen to collect the necessary technical lemmas in this section (to make the first portion of the paper more readable). First, we have a lemma involving L r. Lemma For r and 0 i r, there exist λ r (i) Z such that r L r = λ r (i)h r i, (7) where λ r (i) 0 (mod r i ). (8) Proof. This follows from [5, Lemma ], by observing that K r =L r for the K r defined in [5]. Second, we have a lemma concerning UL r. Lemma For r 5 and 0 i r, there exist δ r (i) Z such that r UL r = δ r (i)h r i, (9) where δ r (i) 0 (mod r i+ ). (0) Proof. This follows from [5, Lemma ]. Third, we prove a crucial lemma about M r. Lemma For r 6, there exist μ r (i) Z such that r M r = μ r (i)h r i, () where and μ r (i) 0 (mod r i ) for 0 i r, () μ r (r ) r (mod r ). () Proof. We use induction on r. By the expression for M 6 in Section 5, we see that the lemma is true for r =6. Suppose that for some r 7 there are integers μ r (j) such that r M r = μ r (j)h r j, () j=0

7 CONGRUENCES MODULO HIGH POWERS OF 6 where and μ r (j) 0 (mod r j ) for 0 j r, (5) μ r (r ) r (mod r ). (6) Then, by () and (5), UM r = = = r μ r (j)uh r j j=0 r j=0 r (r j)/ μ r (j) i k=0 i= j=max(0,i r ) ( ) k r j k ( r j k k ) h r j k ( ) r i ( ) i j r+j i+ μ r (j)h r i. i j Moreover, by (), (7), and (9), so that () holds with M r = L r (UL r ) q + UM r = r r λ r (i)h r i δ r (i)h r i + UM r, μ r (i) =λ r (i) δ r (i) (7) i ( ) r i + ( ) i j r+j i+ μ r (j) i j for 0 i r, and j=max(0,i r ) μ r (r ) = μ r (r ) + μ r (r ). (8) It follows that all the μ r (i) are integers. Furthermore, by (7), (8), (0) and (5), we get (). Finally, () follows from (8), (5) with j = r, and (6). 5 Final Calculations The calculations in this section were performed with the goal of finding a good M r to shape the crucial Lemma, and also to give us a starting point for the induction proof. We had to go all the way up to M 6 before we found an M r which satisfied our requirements.

8 6 ØYSTEIN J. RØDSETH AND JAMES A. SELLERS Our calculations run as follows. (6) (8). We first use (5) to add a few more cases to Uh = h h, Uh = h h + h, Uh 5 = 5 h 5 5 h + h, Uh 6 = 6 h 6 5h 5 + h h Next, we use (0) and the expressions for Uh i to express L r, r 6, in terms of h i. L = 5 h h, L = 9 h 9 h + 5 h, L 5 = h 5 h + 8 9h 7 7h, L 6 = 0 h 6 h 5 + 7h 7h 8 5h Then we apply the U-operator to obtain the following (after elementary simplifications): UL = h h, UL = 8 h 6 h + h, UL = h 5h + 7 h 5 h, UL 5 = 9 h 5 7 7h + h 9 h + 7 7h Finally, we use () and the results above to express M r, r 6, in terms of h i. M = h h h, M = 8 h 6 h, M 5 = h 5 5h + 7 9h + 5 7h, M 6 = 9 h 6 7 7h 5 + 5h + 9 6h 8 5h 5 7h Now, M 6 has the properties we need to prove Lemma. The coefficients of the terms h i are divisible by sufficiently high powers of while the coefficient of h is not divisible by an excessively high power of. This second fact allows us to prove the best possible part of Theorem. From these calculations, we get the following complementary results mentioned in the Introduction for free. The first congruence below follows from (5) and (). For the others, put r =,, 5, 6 in (6) and use the results on L r and M r obtained in the present section. This gives us the following congruences, valid for all n 0: f(n) f(n) n(n +) (mod ), f( n) f(n) n(n +) (mod ), f( n) f( n) 0 (mod 5 ), f( n) f( n) n(n +) (mod 7 ), f( 5 n) f( n) 5 n (mod 8 )

9 CONGRUENCES MODULO HIGH POWERS OF 6 6 Acknowledgements The second author gratefully acknowledges the Department of Mathematics, University of Bergen, Norway for generous support which allowed the two authors to effectively collaborate during a one-week visit to Bergen in March 008. The second author also gratefully acknowledges the staff of the Isaac Newton Institute, University of Cambridge, for their hospitality and support. References [] G.E. Andrews, The Theory of Partitions, Encyclopedia Math. Appl., Vol., Addison-Wesley, Reading, MA, 976. [] R. F. Churchhouse, Congruence properties of the binary partition function, Proc. Cambridge Philos. Soc. 66 (969), [] Ø. J. Rødseth, Some arithmetical properties of m-ary partitions, Proc. Cambridge Philos. Soc. 68 (970), 5. [] Ø. J. Rødseth, Sloane s box stacking problem, Discrete Math. 06 (006), [5] Ø. J. Rødseth and J. A. Sellers, Binary partitions revisited, J. Combin. Theory, Ser. A 98 (00), 5. [6] Ø. J. Rødseth, J. A. Sellers and K. M. Courtright, Arithmetic properties of nonsquashing partitions into distinct parts, Ann. Combin. 8, no. (00), 7 5. [7] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, njas/sequences/ (008). [8] N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math. 9 (005), (Received Apr 008; revised Sep 008)