JOURNAL OF COMBINATORIAL THEORY, Series A 58, 147-152 (1991)

Note

New Modular Properties of Bell Numbers

NABIL KAHALE*

Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Communicated by the Managing Editors Received December 4, 1989; revised March 23, 1990

Given a prime p, we exhibit an c such that the Bell numbers 3 Gil, B,+z> ...> B,+,-, are divisible by p. We calculate the residue of B, modulo P. 0 1991 Academic Press, Inc.

1. INTRODUCTION

The Bell number B, is the number of partitions of a finite set whose cardinality is n. The modular properties of Bell numbers have been studied by many authors. It is well known that the sequence of the residues of Bell numbers modulo any integer satisfies a linear recurrence and is periodic [ 1, 68, 13, 143. When the modulus is prime, this recurrence is particularly simple. Touchard [16] shows the congruence

where p is any . The sequence of residues of B, modulo p is periodic [2, 173 and its period divides T= (pp - 1 )/(p - 1). Radoux [15] conjectures that T is the minimal period of this sequence for any prime p. He shows [14} that if the period of the residues is equal ,to T for a given prime p, then there exists a number c, depending on p, such that B, + 4 SE0 (mod p) for 1~ 4 < p. However, he did not give any general value for c. Radoux’s conjecture has been verified [12] for all primes p less than or equal to 17 but, to our knowledge, it has not been proven in the general case. In Section 3 we show, without assuming any unproven results, the following:

* This research was supported by the Defense Advanced Research Projects Agency under Contract NOOO14-89-J-1988. 147 0097.3165/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. 148 NABIL KAHALE

THEOREM. Let c = (pJ’ - T)/(p - 1). Then c is a positive integer and

B,,+,rO(modp) for l

Furthermore, if p > 2, then

B~,(_1)(P-1)(p-3)/8p-L T! (mod p). (3)

In Section 4, we explore further modular properties of the Bell numbers and show the existence of a sequence of p consecutive Bell numbers having the same residue modulo p.

2. PRELIMINARIES

Throughout this note, p refers to a prime number and 6 is the Kronecker symbol. If a is an element of a ring whose unit element is e, then a” stands for a(a-e)...(a-(n-1)e) and a* for a(u+e)~~~(u+(n--1)e). We say that two infinite integer sequences {u,} and {v,} are congruent modulo p if and only if for all n 3 0, the elements u, and v, have the same residue modulo p. Unless stated otherwise, all congruences are modulo p. From (1) and the recurrence B,, 1 =Ci (1) Bi it is not hard to show [16] that, for all IZ 2 0,

B n+kp= Bn+i, (4)

Bkp--k+l> (5)

B n+ps=@,+Bn+i. (6)

Other properties of the Bell numbers can be found in [3]. De Bruijn [4, p. 1041 gives their asymptotic behavior. The of the second kind, denoted {i}, is defined as the number of partitions of the set { 1, .. . . n} into k blocks. We have

The reader is referred to [11] for other properties of the Stirling numbers. MODULAR PROPERTIES OF BELL NUMBERS 149

3. THE PROOF OF THE THEOREM

Before proving the theorem, we prove the following lemma.

LEMMA. For all n, q > 0,

B n+y- (91 ,’ Proof of lemma. By combining (5) and (6), we have

B,, + pq= qB, + B,,+ I (mod p).

Using (lo), we prove by induction on k that

where E is the shift operator that maps an infinite sequence (un> to the sequence {u, + r >. For the basis, (11) holds for k = 0. Now, assuming that it holds for k, the relation Ek+’ = (E- k) Ek implies that Ea( (B, >) is congruent to {B,, + , jpk - kB,k} which is congruent to (B,@+I} by (10). Thus, (11) holds for k+ 1. Applying both sides of the identity Eq = xi (r} E’ to (B,) and using (11) completes the proof of the lemma. 1 We are now ready to prove the theorem.

Proof of theorem. Since p is congruent to 1, modulo (p - 1 ), so is any power of p. Hence, pJ’ is congruent to 1, modulo (p - I), and so is T = pP-l + . . . + p + 1. Thus pp - T, which is obviously positive, is a multiple of p - 1, and so c is a positive integer. Since pp is congruent to 1, modulo T, so is (p - 1) c. Thus,

pc=c+ 1 (mod T).

Multiplying both sides of (12) by pq yields

CPq+lscpQ+pq (mod T).

Since the period of the residues of B, divides T, (13) shows that BCpu+,z B cp~+pq(mod p). But, using (lo), we have B,,,., = qB+ + B,,+I (mod p). By comparing these two congruences, we see that qB,, is a multiple of p. Since p is prime and q is not a multiple of p, we con&de that BcPqis a mu& tiple of p. An application of the lemma yields the first desired result (2). 150 NABIL KAHALE

Now, we prove (3). Let A, be the p x p matrix 4, &,+I... Bn+p-I

By (1) we have detA,+,=detA,, and thus,

det A, E det A (mod p), (14) where A = A, is the Hankel matrix built on the Bell numbers. But, u! .ng (1) and (2), we see that Br+k= 6 kpB, for 1

B, 0 ... 0 0

. 00:’ ; 0 B, 0 ... 0

Hence, det A, = (- l)(p-1u2 B,P. C ombining this relation with (14) and using Fermat’s theorem yields

(15)

If i lies between 0 and p - 1, then, by [lo, p. 681, we have (P; ‘) - (- 1)’ and therefore i! (p - 1 - i)! E (- l)i (p - l)!. Since, by Wilson’s theorem, (p-1)!r-1,wehavei!(p-1-i)!~(-1)i+1.By[5],however,detA= I-&‘:: k!, and so

detA=p~!‘p~y2i!(p-~-~)!~(-l)‘~-~”p+~’,6P~!. i=O 2

We complete the proof of (3) by replacing det A by its value in (15). g

Remarks. Hardy and Wright [9, p. 881 show that p= 4m + 3 implies ((p - 1)/2)! = ( - 1 )‘, where v is the number of quadratic non-residues less than p/2. In this case, we have B, = ( - l)mf”. If p = 4m + 1, then ((p - 1)/2)! is a square root of - 1, and so B, is also a square root of - 1. MODULARPROPERTIESOFBELLNUMBERS 151

4. OTHER RESULTS

For O

Bc+~+y=Bc+~+q,m (161 B,+,-(q-k)‘%, (171 BckPnE (- l)n k”B,., (181 B,+l-,=O. (191

Congruence (19) holds if s is of the form s = xi=, p”‘, where I> 1 and the nj have distinct residues modulo p. Applying (17) in the particular case k = p - 1 and q = 0 shows us that B cp-,, B+,+l> ...> B+,+p-1 have the same residue modulo p as B,. When p= 5, we have T= 781, c= 586, c,=743, B,,, zzB,,,- Bsg9z BTg, G 0 (mod 5), and B,,, E B,,, z B744E . . . - B,,, = 3 (mod 5).

ACKNOWLEDGMENTS

The author expresses his gratitude to Thomas Cormen, Ira Gessel, and Richard Stanley for their helpful comments and suggestions.

REFERENCES

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