Zhi-Wei Sun (Nanjing)

Transcription

1 Acta Arth , no. 4, COMBINATORIAL CONGRUENCES AND STIRLING NUMBERS Zh-We Sun Nanng Abstract. In ths paper we obtan some sophstcated combnatoral congruences nvolvng bnomal coeffcents and confrm two conectures of the author and Davs. They are closely related to our nvestgaton of the perodcty of the sequence l 0 l S, ma l l m, m + 1,... modulo a prme p, where a and m > 0 are ntegers, and those S, m are Strlng numbers of the second nd. We also gve a new extenson of Glasher s congruence by showng that p 1p log p m s a perod of the sequence r mod p1 l S, m l m, m + 1,... modulo p. 1. Introducton In a recent paper of the author and D. M. Davs [SD] orgnally motvated by the study of homotopy exponents of the specal untary group SUn, the followng sophstcated theorem was establshed. Theorem 1.0 Sun and Davs. Let p be a prme, and let α, n N {0, 1,... } and r Z. Then, for any fx Z[x], we have n r ord p 1 f r mod 1.0 n ord p! deg f + τ p {r} p α1, {n r} p α1, 1 where ord p a sup{m N : p m a} s the p-adc order of a Z, {a} p α1 stands for the least nonnegatve resdue of a modulo 1 and ths s regarded as 0 f α 0, and for a, b N we use τ p a, b to denote the number of carres occurrng n the addton of a and b n base p. Let p be a prme. By a well-nown fact n number theory cf. [IR, p. 26], n ord p n! p for every n 0, 1, 2, Mathematcs Subect Classfcaton. Prmary 11B65; Secondary 05A10, 11A07, 11B73. Supported by the Natonal Scence Fund for Dstngushed Young Scholars Grant No and a Key Program of NSF Grant No n Chna. 1

2 2 Z. W. SUN A useful theorem of E. Kummer asserts that f a, b N then a + b ord p a a + b 1 p a b p p τ p a, b. In ths paper we wll apply Theorem 1.0 to deduce three theorems on combnatoral congruences or Strlng numbers of the second nd. For l, m N wth l + m > 0, the Strlng number Sl, m of the second nd denotes the number of ways to partton a set of cardnalty l nto m nonempty subsets; n addton, we defne S0, 0 to be 1. It s well nown that l x l Sl, x for l 0, 1, 2,..., 0 where x 0 < x and an empty product has the value 1 thus x 0 1. Here s our frst theorem. Theorem 1.1. Let p be any prme. Let a Z, l, l, m Z + {1, 2,... }, l l > m/p and l l mod p 1p log p m δ pa,m, 1.1 where Then we have l 0 { 1 f a pz and logp m Z +, δ p a, m 0 otherwse. l S, ma l l l S, ma l mod p. 1.3 Corollary 1.1. Let p be a prme, and let a Z and m Z +. Then, for m + p 1p log p m δ pa,m q wth q N, we have 0 S, ma 1 mod p. Proof. Just apply Theorem 1.1 wth l m and l. Remar 1.1. Note that f p s a prme and m s a postve nteger then m p 1p log p m < m/p. The followng result was frst obtaned by L. Carltz [C] n See also A. Nenhus and H. S. Wlf [NW], and Y. H. H. Kwong [K].

3 COMBINATORIAL CONGRUENCES AND STIRLING NUMBERS 3 Corollary 1.2. Let p be any prme. Suppose that α, m N, m p and < m +1. Then p 1 s a perod of the sequence {Sl, m} l m modulo p. Proof. It suffces to apply Theorem 1.1 wth a 0. The sum n r mod m wth m Z +, n N and r Z has been nvestgated ntensvely, see [S] for some hstorcal bacground and related congruences. In 1899 J.W.L. Glasher cf. [D, p. 271] and [ST] proved that l l mod p r mod p1 r mod p1 whenever p s a prme, r Z, l, l Z + and l l mod p 1. Clearly Glasher s congruence s our followng result n the case m 1. Corollary 1.3. Let p be a prme, m Z + and r Z. For any l, l Z + wth l l > m/p and l l mod p 1p log m p, we have r mod p1 l S, m r mod p1 l S, m mod p. 1.4 Our second theorem s slghtly stronger than Conecture 1.3 of the author and Davs [SD] whch was proved n [SD] when p 2 and r 0. Theorem 1.2. Let p be a prme, and let α, l, n N and r Z. Set {r}, n + {n r} p α and m n n r n r Suppose that l m > 0 and l m mod p 1p log p m δ p r/, m, 1.6 where the notaton δ p a, m s gven by 1.2. Then we have 1 n/! l n r n 1 1 l+ mod p. 1.7 r mod Remar 1.2. Theorem 1.2 mples that the nequalty n Theorem 5.1 of [DS] s sharp for nfntely many values of l provded that n 2 1. Our thrd theorem confrms Conecture 1.1 of [SD].

4 4 Z. W. SUN Theorem 1.3. Let p be any prme, and let α Z +, l, n N and r Z. Then 1 l pn r n/1 1! p 1 r mod 1 l 1.8 n r n/1 1! 1 mod p a p, where r mod 1 f p 2, a p 2 f p 3, 3 f p > Remar 1.3. Let p be a prme, α, n N and r Z. When > n and l 0 r n, 1.8 reduces to Lunggren s congruence pn pr n r mod p a p cf. [G] whch s an extenson of the Wolstenholme congruence 2p p 2 mod p a p.e., 2p1 p1 1 mod p a p. Note also that 1.8 holds for every l N f and only f we have pn pr for all fx Z[x], where n r f, pdeg f n/1! pα f, +1 r mod n mod p a p 1.10 r f, 1 n f r Z p As usual, Z p denotes the rng of p-adc ntegers. Concernng the rght-hand sde of the congruence 1.8, a Lucas-type congruence modulo p was establshed n [SD] for α > 1 and n [SW] for α 1. See also [SW] for some other congruences of Lucas type related to combnatoral sums nvolvng bnomal coeffcents. In the next secton we are gong to prove Theorem 1.1 and Corollary 1.3. On the bass of Theorem 1.1 we wll deduce Theorem 1.2 n Secton 3. Secton 4 s devoted to our proof of Theorem Proofs of Theorem 1.1 and Corollary 1.3 Proof of Theorem 1.1. By a well-nown property of Strlng numbers of the second nd cf. [LW, pp ], l l l l a l S, m a l 1 m m 1 m m! 0 1 m! 0 m m p1 1 a m + l 1 m S r l, 0 r0

5 COMBINATORIAL CONGRUENCES AND STIRLING NUMBERS 5 where S r l 1 m! r mod p 1 m a + l. 2.1 Let r {0,..., p 1}. Observe that S r l 1 m! l 0 r mod p 1 m l l p a + r m! l 0 r mod p l r p a + r l p m r 1. p By Theorem 1.0, for any N we have σ r : p m! r mod p 1 m r p Z p 2.2 and p m! r mod p m r r 1 p p 0 mod p snce the degree of f x x x Z[x] s smaller than. Therefore, S r l l 0 In vew of the above, t suffces to show that l a + r l σ r l a + r l σ r mod p. 2.3 l a + r l σ r mod p 2.4 for every 0, 1,.... Below we assume N and σ r 0. Then r/p 0 for some 0 m wth r mod p, hence m r r p and m/p < l l. If p a + r, then a + r l 0 a + r l mod p. When p a + r, as l l mod p 1 we have a + r l a + r l mod p

6 6 Z. W. SUN by Fermat s lttle theorem. So t remans to show l l mod p n the case p a + r. Let α log p m. Then m < +1 and δ δ p a, m α. Wrte l δ q 0 + l 0 wth q 0 N and 0 l 0 < δ. For some q N we have l l + p 1δ q δ p 1q + q 0 + l 0. Recall that mr/p <. Suppose a+r 0 mod p. If δ 1, then < m/p 1 because m and r {a} p 0. Thus < δ. Wth help of the Chu-Vandermonde convoluton dentty cf. [GKP, 5.27], l l0 0< 0< δ q 0 δ l0 q 0 δ q l0 0 mod p. Smlarly, l l0 mod p as desred. We are done. Proof of Corollary 1.3. Let g be a prmtve root modulo p. For any nteger h, f p 1 h then p1 a1 ah p 1 1 mod p by Fermat s lttle theorem; f p 1 h then g h 1 mod p and hence p1 a1 ah 0 mod p snce p1 p1 p1 g h 1 a h ag h a h 0 mod p. a1 In vew of the above, Smlarly, p1 a rl a1 l l 0 p1 a rl a1 l 0 l a1 l 0 a1 l S, ma l p1 S, m r mod p1 S, ma l a1 a r l S, m mod p. r mod p1 l S, m mod p. Snce l l mod p1, a rl a rl mod p for all a 1,..., p1. Thus, applyng Theorem 1.1 we mmedately obtan 1.4 from the above.

7 COMBINATORIAL CONGRUENCES AND STIRLING NUMBERS 7 3. Proof of Theorem 1.2 At frst we mae some useful observatons. Clearly m n n r r n r r + < m + 1. Snce { 1 f n, τ p {r}, {n r} τ p {r} p α1, {n r} p α1 0 otherwse, we also have n ord p τ p {r} r p α1, {n r} p α1 n Let a r/, {0,..., n} and r mod. Then r l r l 0 l 0 l a l l a l l + a l 0 n m. 3.1 l a l r r S, 0 m 0 r S,, because for m + 1 we have > n / / and hence / 0. Observe that n n ord p 1! + n n np + ord!. s>α If {0,..., m 1}, then by Theorem 1.0 and 3.1 we have n ord p 1 r mod n ord p 1! + τ p { } 1, {n } 1 n n >ord p! + m + τ p {r} 1, {n r} 1 n n ord p!.

8 8 Z. W. SUN Therefore, 1 n/! n l 0 r mod l S, ma l n/! n 1 n r mod r 1 n l r m mod p. In lght of Corollary 1.1, t remans to show that S 1 l+ mod p, where 1 S n/! n n 1 r. 3.2 m mod If {0,..., n}, mod and / m 0, then m and hence m +. So + S 1mpα n/! n Clearly n m + 0< n n < m + 1 m m 1mpα + m! m + n /! m +n m +r n mpα + n!/mpα +! n!/! / 1 + m pα Thus, f n < then m +n m +r n f n then n / 1 and m +n m + m + 1 n Therefore, 0< n 1 + m pα 0< 1 mod p; / 0< 1 + m pα 1 + m pα S 1 mpα + 1 m+ 1 l+ mod p. Ths concludes the proof of Theorem 1.2. m +n m +r n.. 1 mod p.

9 COMBINATORIAL CONGRUENCES AND STIRLING NUMBERS 9 4. Proof of Theorem 1.3 For, N let δ, be the Kronecer symbol whch taes 1 or 0 accordng to whether or not. Snce δ, 1 1, we have where 1 p1r r mod r mod n pn 1 p p 0 n pn 1 p p 0 n 1 C n, 0 C n, pn r 1 p 1 pn r 1 p p 1 r mod r mod r mod n pn 1 p p 0 As C n,n 1 pn, by the above 1 p1r 1 p1n 0 <n r mod r mod 1 C n, l l r 1 δ, l l r 1 1 l, 1 r mod r 1 0 mod p 1 pn pn r 1 p r mod 1 n 1 1 l l r 1 /p l r 1. Note that 1 p1n 1 p1r mod p a p. In vew of Theorem 1.0, l r ord p 1 1 ord p 1!.. sα

10 10 Z. W. SUN So t suffces to show that ord p C n, a p +ord p n 1! sα a p + sα for any N wth < n. Fx a nonnegatve nteger < n. In lght of Theorem 1.0, pn ord p!c n, ord p p 11! s0 By Lemma 3.2 of [SD] and ts proof, C n, s congruent to 0 n n. n n n n modulo p 2ord pn+a p. In the case p > 3, by Jacobsthal s result cf. [G], f {1,..., n}, then / pn n p for some q Z p, and hence pn p n 1 + p 3 nn q n p 3 nn q p 3 n 2 n 1 n 2 q. 1 So we also have ord p C n, ord p n when p > 3. These facts wll be used n the followng dscusson. Case 1. n a p. In ths case, n n ord p C n, ord p! s0 s0 n n + a p + sα s1 n. Case 2. 0 < n < a p 3, and p n or n 2.

11 COMBINATORIAL CONGRUENCES AND STIRLING NUMBERS 11 If β ord p n > 0, then n < a p < p p β and hence n n/p β n/p β 1 /p β p β+1 p p p p β+1, therefore n sα α s β n α s β 1 < 2β 2ord p n ord p C n, a p. When β ord p n 0.e., p n and n 1, we have sα n 0 2ord p n ord p C n, a p. Case 3. n 2 < a p and p n. In ths case, a p 3 < p and ord p C n, a p ord p n 1 α s ord p n1 n 1 n 2 sα n 1 n 2 sα n. Combnng the above we have completed the proof of Theorem 1.3. References [C] L. Carltz, Congruences for generalzed Bell and Strlng numbers, Due Math. J , [DS] D. M. Davs and Z. W. Sun, A number-theoretc approach to homotopy exponents of SUn, J. Pure Appl. Algebra , [D] L. E. Dcson, Hstory of the Theory of Numbers, Vol. I, AMS Chelsea Publ., [GKP] R. Graham, D. E. Knuth and O. Patashn, Concrete Mathematcs, Addson- Wesley, New Yor, [G] A. Granvlle, Arthmetc propertes of bnomal coeffcents. I. Bnomal coeffcents modulo prme powers, n: Organc mathematcs Burnaby, BC, 1995, , CMS Conf. Proc., 20, Amer. Math. Soc., Provdence, RI, [IR] K. Ireland and M. Rosen, A Classcal Introducton to Modern Number Theory Graduate texts n math.; 84, 2nd ed., Sprnger, New Yor, [K] Y. H. H. Kwong, Mnmum perods of Sn, modulo M, Fbonacc Quart ,

12 12 Z. W. SUN [LW] J. H. van Lnt and R. M. Wlson, A Course n Combnatorcs, 2nd ed., Cambrdge Unv. Press, Cambrdge, [NW] A. Nenhus and H. S. Wlf, Perodctes of partton functons and Strlng numbers modulo p, J. Number Theory , [S] Z. W. Sun, On the sum n r mod m and related congruences, Israel J. Math , [SD] Z. W. Sun and D. M. Davs, Combnatoral congruences modulo prme powers, Trans. Amer. Math. Soc., to appear. [ST] Z. W. Sun and R. Tauraso, Congruences for sums of bnomal coeffcents, J. Number Theory, to appear. [SW] Z. W. Sun and D. Wan, Lucas-type congruences for cyclotomc ψ-coeffcents, Int. J. Number Theory, to appear. Department of Mathematcs Nanng Unversty Nanng People s Republc of Chna zwsun@nu.edu.cn zwsun