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On a Curious Property of Bell Numbers Bull. Aust. Math. Soc. 84 (2011), 153–158 doi:10.1017/S0004972711002218 ON A CURIOUS PROPERTY OF BELL NUMBERS ZHI-WEI SUN ˛ and DON ZAGIER (Received 27 November 2010) Abstract In this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime. 2010 Mathematics subject classification: primary 11B73; secondary 05A15, 05A18, 11A07. Keywords and phrases: Bell numbers, derangement numbers, congruences modulo a prime. 1. Introduction Let Bn denote the nth Bell number, defined as the number of partitions of a set of cardinality n (with B0 D 1). In 1933 Touchard T2U proved that for any prime p we have BpCn ≡ Bn C BnC1 .mod p/ for all n D 0; 1; 2;:::: (1.1) Thus it is natural to look at the numbers Bn .mod p/ for n < p. In T1U, the first author Pp−1 − n discovered experimentally that for a fixed positive integer m the sum nD0 Bn=. m/ modulo a prime p not dividing m is an integer independent of the prime p, a typical case being p−1 X Bn ≡ −1853 .mod p/ for all primes p 6D 2. .−8/n nD0 In this note we will prove this fact and give some related results. Our theorem involves another combinatorial quantity, the derangement number Dn, defined either as the number of fixed-point-free permutations of a set of cardinality n (with D0 D 1) or by the explicit formula n X .−1/k D D nW .n D 0; 1; 2; : : : /: (1.2) n kW kD0 The first author is supported by the National Natural Science Foundation (grant 10871087) and the Overseas Cooperation Fund (grant 10928101) of China. c 2011 Australian Mathematical Publishing Association Inc. 0004-9727/2011 $16.00 153 154 Z.-W. Sun and D. Zagier [2] THEOREM 1.1. For every positive integer m and any prime p not dividing m we have X Bk m−1 ≡ .−1/ Dm− .mod p/: (1.3) .−m/k 1 0<k<p P n−k Using 0<m<p.−m/ ≡ −δn;k .mod p/ for k; n 2 f1;:::; p − 1g, we immediately obtain a dual formula for Bn .n < p/ in terms of D0;:::; Dp−2. COROLLARY 1.2. Let p be any prime. Then for all n D 1;:::; p − 1 we have p−1 n X m n .−1/ Bn ≡ .−1/ m Dm−1 .mod p/: mD1 Combining the case n D p − 1 of this corollary with the congruence p−1 p−1 X X p − 1 .−1/k D ≡ D D .p − 1/W ≡ −1 .mod p/; k k k kD0 kD0 we get the following further consequence of Theorem 1.1. COROLLARY 1.3. For any prime p we have Bp−1 ≡ Dp−1 C 1 .mod p/: For the reader’s convenience we give a small table of values of Bn and Dn. n 0 1 2 3 4 5 6 7 8 Bn 1 1 2 5 15 52 203 877 4140 Dn 1 0 1 2 9 44 265 1854 14833 We will prove Theorem 1.1 in the next section and derive an extension of it in Section3. 2. Proof of Theorem 1.1 We first observe that it suffices to prove (1.3) for 0 < m < p, since both sides are periodic in m .mod p/ with period p. For the left-hand side this is obvious and n for the right-hand side it follows from (1.2), which gives the expression .−1/ Dn ≡ P1 r Q rD0.−1/ 0≤s<r .n − s/.mod p/ for Dn .mod p/ as the sum of a terminating infinite series of polynomials in n. We will prove (1.3) for 0 < m < p by induction on m. Denote by Sm the sum on the left-hand side of (1.3), where we consider the prime p as fixed and omit it from n the notation. Since Dn D nDn−1 C .−1/ for n D 1; 2; 3;::: (obvious from (1.2)), we have to prove the two formulas S1 ≡ 1 .mod p/; mSm ≡ S1 − SmC1 .mod p/: (2.1) [3] On a curious property of Bell numbers 155 Recall that the Bell numbers can be given by the generating function 1 n X x x B D ee −1; (2.2) n nW nD0 equivalent to the well-known closed formula 1 1 X r n B D : n e rW rD0 x Since the function y D ee −1 satisfies y0 D ex y, this also gives the recursive definition n X n B D 1; B C D B for all n ≥ 0: (2.3) 0 n 1 k k kD0 This recursion is the key ingredient in proving (2.1). For the first formula in (2.1) we use (2.3) with n D p − 1 to obtain p−1 p−1 X X p − 1 S D .−1/k B ≡ B D B − B .mod p/; 1 k k k p 0 kD1 kD1 so it suffices to prove that Bp ≡ 2 .mod p/. This is a special case of Touchard’s congruence (1.1), but can also be seen by writing (2.2) in the form 1 X xn X .ex − 1/r x p B D ex C C C O.x pC1/ n nW rW pW nD0 1<r<p to get Bp=pWD 1=pWC (p-integral)C1=pW. Now using Fermat’s little theorem and (2.3), we obtain p−2 n X X n −mS ≡ .−m/p−1−n B m k k nD0 kD0 p−2 p−k−2 X X p − k − 1 ≡ .−1/k B m p−k−1−r .r D n − k/ k r kD0 rD0 p−2 X k p−1−k ≡ .−1/ Bk..m C 1/ − 1/ ≡ SmC1 − S1 .mod p/ kD0 for 1 ≤ m ≤ p − 2. This completes the proof of (2.1) and the theorem. 3. An extension of Theorem 1.1 We now give an extension of Theorem 1.1 from the Bell numbers Bn to the Touchard polynomial Tn.x/, defined by n X k Tn.x/ D S.n; k/x : (3.1) kD0 156 Z.-W. Sun and D. Zagier [4] The numbers S.n; k/ occurring here are the Stirling numbers of the second kind, which can be defined in at least four different ways: • combinatorially as the number of partitions of a set of n elements into k nonempty subsets; • by the generating function 1 X xn .ex − 1/k S.n; k/ D .k ≥ 0/I (3.2) nW kW nD0 • by the recursion relation S.n; k/ D kS.n − 1; k/ C S.n − 1; k − 1/.n; k ≥ 1/ (3.3) together with the initial conditions S.n; 0/ D S.0; n/ D δn;0; • by the closed formula k − X .−1/k j jn S.n; k/ D : (3.4) jW.k − j/W jD0 From the first or third of these we see that Tn.1/ D Bn, so the following result includes Theorem 1.1 as the special case x D 1. THEOREM 3.1. For any prime number p and any positive integer m not divisible by p, m−1 m X Tn.x/ p X .m − 1/W l .−x/ ≡ −x .−x/ .mod pZpTxU/; (3.5) .−m/n lW 0<n<p lD0 where Zp denotes the ring of p-adic integers. Observe that this congruence of polynomials implies the congruence m−1 X Tn.x/ 1 X .m − 1/W ≡ .−x/l .mod p/ .−m/n −x m−1 lW 0<n<p . / lD0 for any p-adic integer x not divisible by p, special cases being X Tn.x/ x − 1 ≡ .mod p/ for p 6D 2; .−2/n x 0<n<p 2 X Tn.x/ x − 2x C 2 ≡ .mod p/ for p 6D 3; .−3/n x2 0<n<p 3 2 X Tn.x/ x − 3x C 6x − 6 ≡ .mod p/ for p 6D 2: .−4/n x3 0<n<p PROOF OF THEOREM 3.1. One can prove Theorem 3.1 by a slight modification of the argument used for Theorem 1.1, replacing the recursion (2.3) for the Bell numbers by [5] On a curious property of Bell numbers 157 the analogous recursion n X n T .x/ D 1; T C .x/ D x T .x/ for n ≥ 0 0 n 1 k k kD0 for the Touchard polynomials. But it is in fact easier to prove this congruence of polynomials for each coefficient separately. We first observe that, just as in the case of Theorem 1.1, it suffices to consider m among 1;:::; p − 1, since by setting l D m − 1 − r we can rewrite (3.5) in the form 1 X Tn.x/ X Y p−1−r −1 ≡ .m − s/ .−x/ .mod pZpTxUTTx UU/ .−m/n 0<n<p rD0 1≤s≤r in which both sides depend only on m .mod p/. (Note that the expression on the Q right is a polynomial modulo p, since 1≤s≤r .m − s/ is divisible by p for r ≥ p.) Comparing the coefficients of xk on both sides of this identity, we see that it suffices to prove the congruence p−1 X S.n; k/ Y ≡ .s − m/.mod p/ (3.6) .−m/n nDk 0<s<p−k for m and k in f1;:::; p − 1g. We can do this in two different ways.
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