ANA BERRIZBEITIA, LUIS A. MEDINA, ALEXANDER C. MOLL, VICTOR H. MOLL, AND LAINE NOBLE

THE p-adic VALUATION OF STIRLING NUMBERS ANA BERRIZBEITIA, LUIS A. MEDINA, ALEXANDER C. MOLL, VICTOR H. MOLL, AND LAINE NOBLE Abstact. Let p > 2 be a pime. The p-adic valuation of Stiling numbes of the second kind is analyzed. Two types of tee diagams that encode this infomation ae intoduced. Conditions that descibe the infinite banching of these tees, simila to the case p = 2, ae pesented. 1. Intoduction The Stiling numbes of second kind S(n, k), defined fo n N and 0 k n count the numbe of ways to patition a set of n elements into exactly k nonempty subsets. They ae explicitly given by (1.1) S(n, k) = 1 k 1 ( 1) i (k i) n, k! i o, altenatively, by the ecuence (1.2) S(n, k) = S(n 1, k 1) + ks(n 1, k), i=0 with the initial conditions S(0, 0) = 1 and S(n, 0) = 0 fo n > 0. Divisibility popeties of intege sequences ae expessed in tems of p-adic valuations: given a pime p and a positive intege b, thee exist unique integes a, n, with a not divisible by p and n 0, such that b = ap n. The numbe n is called the p adic valuation of b. We wite n = ν p (b). Thus, ν p (b) is the highest powe of p that divides b. The 2-adic valuation of the Stiling numbes was discussed in [1]. The main conjectue descibed thee is that the patitions of the positive integes N in classes of the fom (1.3) C m,j := {2 m i + j : i N} whee the index i stats at the point whee 2 m i + j k, leads to a clea patten fo the 2-adic valuations of Stiling numbes. The class C m,j is called constant if ν 2 (C m,j ) consists of a single value. This single value is called the constant of the class C m,j. The paamete m in (1.3) is called the level of the class. The level ae defined inductively as follows: assume that the (m 1)-level has been defined and that it consists of the s classes (1.4) C m 1,i1, C m 1,i2,..., C m 1,is. Date: August 31, 2009. 2000 Mathematics Subject Classification. Pimay 11B50, Seconday 05A15. Key wods and phases. valuations, Stiling numbes. 1

2 A. BERRIZBEITIA ET AL. Each class C m 1,ij splits into two classes modulo 2 m, namely C m,ij and C m,ij+2 m 1. The m-level is fomed by the non-constant classes modulo 2 m. The main conjectue of [1] is now stated: Conjectue 1.1. Let k N be fixed. Then we conjectue that a) thee exists a level m 0 (k) and an intege µ(k) such that fo any m m 0 (k), the numbe of nonconstant classes of level m is µ(k), independently of m; b) moeove, fo each m m 0 (k), each of the µ(k) nonconstant classes splits into one constant and one nonconstant subclass. The latte geneates the next level set. This conjectue was established in [1] only fo the case k = 5. The poof makes stong use of the ecuence (1.2) and the fact that the 2-adic valuations fo S(n, k), fo 1 k 4 ae easily detemined. The goal of this pape is to descibe a simila behavio fo the p-adic valuations of S(n, k) fo the case of p an odd pime. 2. Modula and p-adic tees Conside a sequence of positive integes a := {a j }, with a j dividing a j+1. The tee associated to a stats with a oot vetex and as fist geneation has a 1 vetices epesenting all the esidue classes modulo a 1. Each vetex, banches into the second geneation into a 2 /a 1 siblings coesponding to esidues modulo a 2. Fo instance, the vetex coesponding to 1 in the fist geneation banches into the vetices coesponding to the classes 1, 1 + a 1, 1 + 2a 1,...,1 + (a 2 /a 1 1)a 1. The m-th geneation of the tee contains a m vetices that povide a patition of N into a collection of esidue classes. We define two types of tees that encode the p-adic valuations of the Stiling numbes. In each case, the sequence a j descibed above coesponds to the peiod of the function S(n, k) modulo powes of the pime p. This peiodicity is descibed fist. Given a pime p and an intege m N, the sequence S(n, k) mod p m, fo k fixed is peiodic of peiod L m := (p 1)p m+α(p,k) 1, whee α(p, k) is defined by p α(p,k) < k p α(p,k)+1. Obseve that L m divides L m+1 with quotient p. Fo instance, the Stiling numbes S(n, 5) modulo 3 2 have peiod 18, the epeating block being (2.1) {1, 6, 5, 6, 3, 0, 4, 6, 8, 6, 6, 0, 7, 6, 2, 6, 0, 0}. The question of the minimal peiod of S(n, k) modulo p m has been discussed by Nijenhuis and Wilf [5] and Kwong [4]. This pape is not concened with minimality and uses only the peiod L m defined above. See [2], [3] fo moe infomation on this topic. Indexing. Fix a pime p and m N. The esidue classes modulo L m+1 can be witten in the fom (2.2) n = i 0 + i 1 L 1 + i 2 L 2 + + i m L m, whee 0 i 0 p 2 = L 1 1 and 0 i p 1 fo 1 m. The coefficients i ae uniquely detemined by n.

p-adic VALUATION 3 Definition 2.1. The set (2.3) D m,j := {n N : n j mod L m }, will be called a class at the m-th level. The tee associated to the sequence {L j } is now modified by banching a subset of the nodes at the m-th level. Recall that each vetex at the m-th level coesponds to a class D m,j. Two diffeent banching citeia poduce the two types of tees mentioned above. Note. We assume thoughout that k < p. This has the advantage that the tem k! in (1.1) is invetible modulo p, so it may be ignoed in the question of divisibility of S(n, k) by p. This assumption also yields α(p, k) = 0 and the sequence S(n, k) mod p m is peiodic of peiod L m := (p 1)p m 1 = ϕ(p m ), whee ϕ is the Eule totient function. Modula tees. A banching vetex is one fo which (2.4) S(D m,j, k) 0 mod p m. The emaining vetices will be called teminal. The peiodicity of S(n, k) modulo p m shows that (2.4) is independent of the element in the class D m,j. Indeed, if n 1 n 2 mod p 1, then (2.5) S(n 1, k) S(n 2, k) mod p. In paticula, if S(n 1, k) 0 mod p, then ν p (S(n, k)) = 0 fo all n n 1 mod (p 1). On the othe hand, if S(n 1, k) 0 mod p, then is not guaanteed that ν p (S(n 1, k)) and ν p (S(n 2, k)) ae the same. 0 0 Figue 1. The fist level fo p = 5 and k = 3. The fist level consists of p 1 vetices coesponding to the esidue classes modulo L 1 = p 1. The banches that coespond to values with S(n, k) 0 mod p ae labeled by the valuation, namely 0, they ae teminated. The emaining banches ae not labeled and ae split modulo (p 1)p into the next level. The p edges coming out of a banching vetex i coespond to the p numbes n j := i + (p 1)j with 0 j p 1. All these indices ae conguent modulo p 1 and distinct modulo L 2 := p(p 1). Thei coesponding Stiling numbes

4 A. BERRIZBEITIA ET AL. 0 0 1 1 1 1 Figue 2. The fist and second level fo p = 5 and k = 3. satisfy ν p (S(n j, k)) 1. At this point, we conside the values of S(n j, k) modulo p 2. Using the fact that (2.6) a b mod L 2 implies S(a, k) S(b, k) mod p 2, we conclude that, if S(n j, k) 0 mod p 2, then ν p (S(n j, k)) = 1. Those vetices ae labeled 1 and teminated. The emaining vetices, namely those fo which S(n j, k) 0 mod p, ae split again modulo p. Fo these classes we have that thei valuations ae at least 2. Example 2.2. Figues 1 and 2 illustate the fist two levels fo p = 5 and k = 3. At the fist level, the integes ae divided in classes modulo L 1 = 4 and the coesponding Stiling numbes ae consideed modulo p = 5. Figue 1 shows that two classes, those conguent to 0 and 3 modulo 4, have Stiling numbes not divisible by 5. Thus, ν 5 (D 1,0 ) = ν 5 (D 1,3 ) = 0. These banches ae labeled by thei valuation. The Stiling numbes coesponding to the two emaining classes do not have the same valuation. At this point one can only conclude that ν 5 (n, 3) 1 fo n 1 o 2 mod 4. Fo example, ν 5 (S(13, 3)) = ν 5 (7508501) = 3 ν 5 (S(17, 3)) = ν 5 (5652751651) = 2. The numbes conguent to 1 modulo 4 split into classes conguent to 1, 5, 9, 13, 17 modulo L 2 = 20 = 4 5. Similaly, the class 2 modulo 4 becomes the five classes 2, 6, 10, 14, 18 modulo 20. The constuction of the tee continues by testing the Stiling numbes coesponding to these indices modulo 5 2 = 25. Lemma 2.3. If a vetex j appeas at the level m, then ν p (D m,j ) m 1. The modula tee has the advantage that it is easy to geneate: at the m-th level the Stiling numbes ae tested modulo p m. Moeove, if a banch teminates at level m, then the coesponding indices satisfy ν p (n) = m. Expeimentally it is found that they become vey lage. This is a disadvantage. An altenative to these tees is poposed next. p-adic tees. This second type of tees diffe fom the modula tees in the mechanism employed to teminate a banch. Recall that, at level m, each vetex coesponds to a class D m,j. In this new type of tees, a vetex is declaed teminal if the coesponding class has constant p-adic valuation; i.e. fo n D m,j, the valuation ν p (S(n, k)) is independent of n. In pactice, this is decided in an expeimental

p-adic VALUATION 5 manne. Given a paamete T, the class D m,j is declaed constant if its fist T -values have the same valuation. Natually a class is banched if it contains two elements with distinct valuation. The last section pesents an algoithm whee this expeimental pocedue is eplaced by a igoous pocedue. 0 0 1111 22222 22222 22222 2222 Figue 3. The fist thee levels of the modula tee fo p = 5 and k = 3 0 0 2 2 2 1 1 1 1 3 3 3 3 3 3 3 3 2 2 2 2 Figue 4. The fist thee levels of the p-adic tee fo p = 5 and k = 3 Figues 3 and 4 pesent the fist thee levels of the modula and 5-adic tees fo k = 3, espectively. Obseve that in the modula tees thee ae vetices that banch to the thid level only to discove that each of the siblings has the same constant valuation. This is the case fo the second vetex at the second level. The p-adic tee detects this phenomena at the second level. This accounts fo the fact that, in geneal, p-adic tees ae smalle that the modula tees. The study of the p-adic valuations of Stiling numbes begins with an elementay citeia. Lemma 2.4. Let i be a esidue class modulo L 1 such that S(i, k) 0 mod p. Then i is a teminal vetex and the class D 1,i has p-adic valuation 0. Example 2.5. Take p = 7 and k = 4. The peiodicity of the Stiling numbes shows that if i 0 mod L 1 = 6, then S(i, 4) S(6, 4) mod 7. In this case S(6, 4) =

6 A. BERRIZBEITIA ET AL. 65 2 mod 7, thus S(6t, 4) 2 mod 7. Theefoe the class D 1,0 has constant 7-adic valuation 0 and the vetex 0 is teminal. 0 0 0 Figue 5. The fist level fo p = 7 and k = 4 Figue 5 shows that the vetices 0, 4 and 5 ae teminal and the classes D 1,0, D 1,4 and D 1,5 have constant valuation 0. The emaining classes D 1,1, D 1,2 and D 1,3 ae now tested futhe. The desciption is given fo the class D 1,1, the othes ae teated using the same ideas. Details ae given in section 5. Fo level m = 1 and if n 1 mod L 1 = 6, we have (2.7) S(1 + 6t, 4) S(L 1 + 1, 4) = S(7, 4) mod 7. The value S(7, 4) = 350, shows that S(7, 4) 0 mod 7. We conclude that the valuation ν 7 (S(1+6t, 4)) 1. The question of whethe the class D 1,1 is constant is decided in an expeimental manne. The equied numbe of values to declae the banching of a class is usually small. Fo instance, fo D 1,1, the value T = 7 suffices in pactice. Indeed, S(43, 4) = 7 2 N, with N = 65790764819319273461750 0 mod 7. Theefoe, the class D 1,1 is not constant. The value T = 7 is also sufficient to detemine that the two emaining classes, D 1,2 and D 1,3 ae nonconstant. The analysis of the banched class D 1,1 is now consideed at level 2, that is, the sequence n = 1+6t is consideed modulo L 2 = (p 1)p = 42. The class D 1,1 splits into the seven classes (2.8) D 2,1, D 2,7, D 2,13, D 2,19, D 2,25, D 2,31, D 2,37. The banches of the tee depicted in Figue 6 show the banching fom the nonconstant class D 1,1. Recall the indexing fo esidue classes modulo L 2 = 42 in the fom n = i 0 + i 1 L 1 descibed in (2.2). The subtee depicted in Figue 6 coesponds to indices with i 0 = 1 and the label in each banch is the esidue of S(i 0 + i 1 L 1 + tl 2, 4) modulo 7 2. Note. Figue 6 shows that six of the seven classes in (2.8) ae constant. This leaves only the leftmost vetex, with label i 0 = 1, i 1 = 0, fo possible banching. This is the class (2.9) D 2,1 = {n N : n 1 mod 42} = {1 + 42t : t N}.

p-adic VALUATION 7 D 1,1 1,0 7,7 13,14 19,21 25,28 31,35 37,42 Figue 6. Banching into the second level 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Figue 7. The second level fo p = 7 and k = 4 Then one checks that D 2,1 banches to level 3 by computing the values of ν 7 (S(1+ 42t, 4)) fo 0 t T. Once again, T = 7 is sufficient to veify that D 2,1 is a non-constant class. Note. In ode to declae a class D m,j constant, we compute the fist T values of ν p (S(j +tl m, k)) fo a pecibed length T. The class D m,j is consideed constant if these values ae the same. Expeiments have shown that a modest value T suffices in each case. 3. Citeia fo banching on modula tees The goal of this section is to discuss the banching mechanism on modula tees. The assumption k < p and the labeling descibed in (2.2) ae imposed thoughout. Recall that each vetex at level m epesents a class D m,j = {n N : n j mod L m }. Using the value L 1 = p 1, it follows that level 1 is epesented by

8 A. BERRIZBEITIA ET AL. the numbes {0, 1, 2,..., p 2} and a vetex i 0 banches to level 2 if (3.1) S(i 0 + tl 1, k) 0 mod p. The teminal (= non-banching) vetices stop at level 1 and they ae labeled by the valuation ( = 0). The banched vetices ae now consideed modulo L 2 and they banch to level 3 if (3.2) S(i 0 + i 1 L 1 + tl 2, k) 0 mod p 2. Continuing with this pocess, the vetices appeaing at level m coespond to banches that have not been teminated up to this point. Theefoe the coesponding classes satisfy ν p (D m,j ) m 1. Lemma 3.1. Assume the a vetex i at level m satisfies (3.3) S(i, k) 0 mod p m. Then ν p (D m,i ) = m 1 and the vetex is teminal. The fist esult states that the left most vetex always teminates. Lemma 3.2. Assume n 0 mod L 1. Then S(n, k) 0 mod p and ν p (S(n, k)) = 0. Theefoe the vetex D 1,0 is teminal. Poof. Wite n = i(p 1) and then (3.4) S(n, k) = ( 1)k k! ( 1) i(p 1). Femat s little theoem shows that p 1 1 mod p, thus ( ) (3.5) S(n, k) ( 1)k k ( 1) mod p. k! The esult now follows fom (3.6) ( 1) = 1. Definition 3.3. Assume 1 p 1, so that L1 1 mod p. Define β by the identity L1 1 + pβ mod p 2. The distibution of β is complicated. Figue 8 shows the values of β fo 2 7918. The pime 7919 is the 1000-th pime. The expession β is now employed to compute the esidue of S(n, k) modulo p 2. Theoem 3.4. Let n i 0 + i 1 L 1 mod L 2. Then [ ( ) (3.7) S(n, k) ( 1)k k ( ] k ( 1) i0 + i 1 p ( 1) ) i0 β mod p 2. k!

p-adic VALUATION 9 8000 6000 4000 2000 2000 4000 6000 8000 Figue 8. The values of β fo the pime p = 7919. Poof. The elation L2 1 mod p 2 yields ( ) (3.8) S(n, k) ( 1)k k ( 1) i0 i1l1 mod p 2. k! Theefoe S(n, k) ( 1)k k! ( 1)k k! ( 1)k k! ( 1)k k! This gives the esult. ( 1) i0 i1l1 mod p 2 ( 1) i0 (1 + pβ ) i1 mod p 2 ( 1) i0 (1 + i 1 pβ ) mod p 2 ( k ( 1) ) i0 + ( 1)k i 1 p k! ( 1) i0 β mod p 2. Coollay 3.5. Let n i 0 mod L 1. Then ( ) (3.9) S(n, k) ( 1)k k ( 1) i0 mod p. k! Theefoe the class D 1,i0 banches to level 2 pecisely when (3.10) ( 1) i0 0 mod p. Coollay 3.6. Let n i 0 mod L 1 with 1 i 0 k 1. Then the class D 1,i0 banches to level 2. Poof. The esult follows fom the pevious coollay and (3.11) ( 1) i0 = ( 1) k k!s(i 0, k) = 0

10 A. BERRIZBEITIA ET AL. in view of the assumption 1 i 0 k 1. Coollay 3.7. Let D 2,j be a class at level 2 with j i 0 + i 1 L 1 mod L 2. Then D 2,j banches to level 3 pecisely when (3.12) ( 1) i0 0 mod p 2 and (3.13) i 1 ( 1) i0 β 0 mod p. Coollay 3.8. Let D 2,j be a class at level 2 with j i 0 + i 1 L 1 mod L 2. Assume 1 i 0 k 1. Then D 2,j banches to level 3 pecisely when (3.14) i 1 ( 1) i0 β 0 mod p. In paticula, the class coesponding to i 1 = 0 always banches. The emainde of this section addesses pattens of banching behavio that continue though inceasing m-levels. Lemma 3.9. Fo given p and 1 < p, fo all m, with β independent of m. ϕ(pm) 1 + β p m mod p m+1 Poof. Suppose ϕ(pm 1) 1 + β p m 1 mod p m fo some m. Then This is the esult. ϕ(pm ) = ϕ(pm 1 )p (1 + β p m 1 + ap m ) p mod p m+1 1 + β p m mod p m+1. Now conside a banching class D m,j at level m. Its siblings consist of the p classes D m+1,j+ilm with 0 i p 1. The next esult states a condition that guaantees the pesevation of the paiwise inconguent popety. In paticula, fom some point on, exactly one of them will banch. Theoem 3.10. Let m 1. Assume that fo some banching class D m,j all vetices ae inconguent, that is, (3.15) S(D m+1,j+i1l m, k) S(D m+1,j+i2l m, k) mod p m+1 fo 0 i 1 < i 2 p 1. Then, evey subsequent banching class D M,J with J j mod L m and M m, (3.16) S(D M+1,J+i1L M, k) S(D M+1,J+i2L M, k) mod p M+1 fo 0 i 1 < i 2 p 1.

p-adic VALUATION 11 Poof. Let d 1, d 2 D m,j such that d 1 d 2 mod L m+1. Index these values by d 1 = j + i 1 L m + x 1 L m+1, d 2 = j + i 2 L m + x 2 L m+1, with x 1, x 2 Z. By assumption, S(d 2, k) S(d 1, k) 0 mod p m+1. Then ( 1) k k! [S(d 2, k) S(d 1, k)] ( 1) [ d2 d1 ] mod p m+1 ( 1) j ( Lm ) i1 [( Lm ) i2 i1 1] mod p m+1 ( 1) j (1 + β p m ) i1 [(1 + β p m ) i2 i1 1] mod p m+1 (i 2 i 1 )p m ( 1) j β mod p m+1. Obseve that the hypothesis of paiwise inconguence imply that and convesely. ( 1) j β 0 mod p Since the Stiling numbes associated with the p vetices emanating fom D m,j ae by hypothesis paiwise inconguent modulo p m+1, exactly one of these vetices banches. Suppose that this banching vetex coesponds to the class D m+1,j+i L m. Now wite d 1 = j + i L m + i 1 L m+1 + x 1 L m+2 d 2 = j + i L m + i 2L m+1 + x 2 L m+2 whee again 0 i 1 < i 2 p 1. To veify that the paiwise inconguence condition holds at this new level, it is equied to show that (3.14) S(d 2, k) S(d 1, k) 0 mod p m+2. Poceeding as befoe, we have ( 1) k k! [S(d 2, k) S(d 1, k)] Expanding as befoe, it follows that ( k [ ( 1) ) d 2 d1] mod p m+2. ( 1) k k! [S(d 2, k) S(d 1, k)] (i 2 i 1 )pm+1 ( 1) j+i L m β mod p m+2. The assumption (3.13) poves the paiwise inconguence of the Stiling numbes at level m + 2.

12 A. BERRIZBEITIA ET AL. Coollay 3.11. Assume the class D m,j has satisfies the paiwise inconguence condition at level m. Then it satisfies at level m + 1 if and only if (3.13) ( 1) j β 0 mod p. holds. Obseve that this condition is independent of m. Note. A banching class D m,j satisfying condition (3.13) sepaates at level m + 1 into p 1 teminal vetices (with Stiling numbes of constant p-adic valuation m) and exactly one banching vetex. Obseve that this is exactly the analog of the main conjectue of [1]. Note. Let j be an index such that the hypothesis of Theoem 3.10 is satisfied fo D m,j at level m. Wite (3.14) j = i 0 + i 1 L 1 + + i m 1 L m 1, and obseve that m 1 j = i0 t=1 Ltit i0 mod p. Thus, the condition (3.13) depends only on j modulo L 1. Coollay 3.12. Using the expansion (3.15) j = i 0 + i 1 L 1 + + i m 1 L m 1, the hypothesis of Theoem 3.10 is satisfied fo D m,j at level m if and only if it is satisfied fo D 1,i0. Coollay 3.13. If (3.13) is not satisfied, that is, (3.16) ( 1) j β 0 mod p, fo j, then fo d 1, d 2 such that d 1 d 2 j mod L 1 and d 1 d 2 mod L m, S(d 1, k) S(d 2, k) mod p m+1. Example 3.14. This example continues Example 2.2 fo p = 5 and k = 3. Conguence (3.16) is satisfied fo the class D 1,1, that is, fo n 1 mod L 1 = 4. Since 29 49 1 mod 4 and 29 49 mod 20, Coollay 3.13 shows that S(29, 3) S(49, 3) mod 5 3. This is depicted in Figue 9. Moeove, the class D 2,9 coesponds to a teminal vetex: fo all n such that n 9 mod 20, ν 5 (S(n, 3)) = 2. 4. Refined banching citeia fo modula tees In this section we descibe a efinement of the citeia developed in the pevious section. Examples illustating the use of this citeia ae given in Section 5. The goal is to povide a genealization of Theoem 3.10. We stat by genealizing Definition 3.3. Definition 4.1. Fix a pime p. Given 1 < p, define the sequence β 1,, β 2,,..., β ω+1, by (4.1) ϕ(pω+1) 1 + β 1, p ω+1 + β 2, p ω+2 +... + β ω+1, p 2ω+1 mod p 2ω+2.

p-adic VALUATION 13 D 2,9 0,25 1,25 2,25 3,25 4,25 D 3,9 D 3,29 D 3,49 D 3,69 D 3,89 Figue 9. Banching fom the class D 2,9. S(9 + i 1 20, k) S(9 + i 2 20, k) mod 5 3, fo all i 1, i 2. Next a genealization of Lemma 3.9. Lemma 4.2. The coefficients β 1,, β 2,,...,β ω, ae independent on ω i.e. if m > ω then (4.2) ϕ(m) 1 + β 1, p m + β 2, p m+1 + + β ω, p ω+m 1 mod p ω+m Poof. The poof is simila to the one fo Lemma 3.9. Definition 4.3. Define the condition (4.3) T ω+1,j := ( 1) j (β 1, + β 2, p +... + β ω+1, p ω ) 0 mod p ω+1 fo j modulo L ω+1, ω 0. Obseve that T 1,j coincides with (3.13). Poposition 4.4. T ω+1,j tue (o false) implies T ω+1,j+iml m tue (o false) fo all m ω + 1. Poof. Fo 1 < p, Lm 1 mod p m, so Lm 1 mod p ω+1 fo m ω + 1. Just as the condition T 1,j seved as a test fo a paticula banching behavio continuing though m-levels, the collection of conditions T 1,j, T 2,j,..., T ω,j now seves as a test. The implication of these conditions is pesented in the following poposition and theoem. Poposition 4.5. Conside a banching class D j,m. Suppose that T j,m is false, then p 2m divides S(j + il m, k) fo all i N pecisely when (4.4) ( 1) j 0 mod p 2m.

14 A. BERRIZBEITIA ET AL. Poof. Note that ( 1) k k!s(j + al m, k) ( 1) j mod p 2m +a ( 1) j ( β 1, p m + β 2, p m+1 + + β n, p 2m 1) mod p 2m. Since T j,m is false, the second sum is conguent to zeo modulo p 2m. The esult is established. Theoem 4.6. Suppose that D m,j is a banching class. If T 1,j, T 2,j,...,T m 1,j ae false and T m,j is tue, then all siblings of D m,j at level m + 1 ae inconguent modulo p 2m. Moeove, all siblings of evey subsequent banching class D M,J with J j mod L m and M > m ae inconguent modulo p m+m. Poof. The poof of this theoem is simila to the one of Theoem 3.10. Let d 1, d 2 D m,j such that d 1 d 2 mod L m+1. Index these values by whee i 1 < i 2 and x 1, x 2 Z. Note that d 1 = j + i 1 L m + x 1 L m+1, d 2 = j + i 2 L m + x 2 L m+1 (4.5) ( 1) k k![s(d 1, k) S(d 2, k)] (i 2 i 1 )p m T m,j mod p 2m. Hee we ae abusing the notation and letting (4.6) T m,j = ( 1) j ( β 1, + β 2, p 1 + + β n, p m 1) athe than the test itself. By assumption T m,j 0 mod p m, hence (4.5) is not conguent to zeo modulo p 2m. The est of the poof follows by induction as in Theoem 3.10. Example 4.7. Theoem 4.6 descibes the behavio of the class D 2,11 fo p = 5 and k = 4. T 1,11 is false, but T 2,11 is tue. Then since 231 391 11 mod 20 and 231 391 mod 100, S(231, 4) S(391, 4) mod 5 4. This is depicted in Figue 10. 5. The p-adic valuation of Stiling numbes In this section we pesent thee illustative examples of the algoithm developed in this pape. We pesent the data using p-adic tees, since they do not gow as fast as modula tees. A conjectue on the stuctue of these valuations is also pesented. This conjectue extends the esults pesented in [1]. Recall the constuction of tees descibed hee: fix a pime p and an index k satisfying 0 k < p. The Stiling numbes S(n, k) ae peiodic modulo p m, of peiod L m = (p 1)p m 1. The desciption of the pattens of the valuations ν p (S(n, k)) is given in tems of the classes (5.1) D m,j = {n N : n j mod L m }.

p-adic VALUATION 15 D 2,11 0 0 0 0 0 D 3,11 D3,31 D 3,51 D 3,71 D 3,91 125 125 125 125 125 375 375 375 375 375 0 0 0 0 0 250 250 250 250 250 500 500 500 500 500 Figue 10. Banching fom the class D 2,11, fo p = 5 and k = 4. Each class coesponds to a vetex of a tee and, fo each m N, the collection of all classes D m,j, is called the m-th level of the tee. The fundamental connection between classes and divisibility popeties is given by (5.2) a, b D m,j if and only if S(a, k) S(b, k) mod p m. Example 5.1. Let p = 7 and k = 3. Figue 11 pesents the p-adic tee coesponding to these numbes. 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 Figue 11. The 7-adic tee of S(n, 3) The fist level of the tee associated with this case coesponds to the esidues modulo L 1 = 6. This gives the six classes D 1,0, D 1,1, D 1,2, D 1,3, D 1,4 and D 1,5 that fom the fist level. Lemma 3.2 guaantees that the class (5.3) D 1,0 = {n N : n 0 mod 6} teminates and it has valuation 0. On the othe hand, the classes (5.4) (5.5) (5.6) D 1,3 = {n N : n 3 mod 6} D 1,4 = {n N : n 4 mod 6} D 1,5 = {n N : n 5 mod 6}

16 A. BERRIZBEITIA ET AL. also teminate with valuation 0. The eason is simple: the class D 1,3 contains 3 and S(3, 3) = 1. Peiodicity of S(n, 3) implies that evey index in D 1,3 satisfies S(4t + 3, 3) 1 mod 7. Theefoe ν 5 (D 1,3 ) = 0. Similaly, D 1,4 contains 4 and S(4, 3) = 6 and D 1,5 contains 5 and S(5, 3) = 25. The emaining two classes, D 1,1 and D 1,2, satisfy S(D 1,1, k) S(D 1,2, k) 0 mod 7, hence testing is equied. The D 1,1 class has a simple banching stuctue. At level 2 D 1,1 has seven siblings D 2,1, D 2,7, D 2,13, D 2,19, D 2,25, D 2,31 and D 2,37. These classes ae labeled by the index i 0 + i 1 L 1 modulo L 2 = 42. The label i 0 = 1 and 0 i 1 6. A diect calculation shows that S(2 + 0 6 + tl 2, 3) 0 mod 7 2 S(2 + 1 6 + tl 2, 3) 7 mod 7 2 S(2 + 2 6 + tl 2, 3) 14 mod 7 2 S(2 + 3 6 + tl 2, 3) 21 mod 7 2 S(2 + 4 6 + tl 2, 3) 28 mod 7 2. S(2 + 5 6 + tl 2, 3) 35 mod 7 2 S(2 + 6 6 + tl 2, 3) 42 mod 7 2. Obseve that these values ae inconguent modulo 7 2. An altenative way to see this is by noticing that condition (3.13) holds 3 ( ) 3 (5.7) ( 1) β 1 mod 7. Theefoe, Theoem 3.10 states that this patten of paiwise inconguence will continue though inceasing m-levels, as illustated in Figue 11. The class D 1,2 also has a simple banching stuctue since a simila agument poduces 3 ( ) 3 (5.8) ( 1) 2 β 5 mod 7. It follows that each of the classes D 1,1 and D 1,2 split modulo L 2 = 42 into six teminal classes, with constant ν 7 (S(n, 3)) = 1, and one banching (nonconstant) class. This banching class will moeove split modulo L 3 = 294 into six teminal classes with constant ν 7 (S(n, 3)) = 2 and one banching class, and so on, at each m-level. This ends the chaacteization of this tee. Example 5.2. The next example consides the valuations ν 5 (S(n, 4); that is, p = 5 and k = 4, see Figue 12. The fist level is fomed by the classes D 1,0, D 1,1, D 1,2 and D 1,3. Lemma 3.2 states that the class D 1,0 is teminal with valuation ν 5 (D 1,0 ) = 0. The emaining thee classes ae subject to the pe-test: is S(j + 4t, 4) 0 mod 5? If the answe is no, the vetex j is teminal and ν 5 (D 1,j ) = 0. Fom j k 1 = 3, Theoem 3.6 shows that S(j + 4t, 4) 0 mod 5 fo 1 j 3 and t N. In ode to decide if these vetices banch to level 3, we ask the question: does T 1,j hold?, that is, is 4 (5.9) T 1,j := ( 1) j β 1, 0 mod 5?

p-adic VALUATION 17 0 1111 1111 2 22 2222 2222 3333 33 33 3333 3333 4444 44 44 4444 4444 5555 55 55 Figue 12. The 5-adic tee of S(n, 4) Recall that β is defined in (3.3). A diect calculation shows that (5.10) β 1,1 = 0, β 1,2 = 3, β 1,3 = 1, β 1,4 = 1 and this gives that T 1,1 and T 1,2 ae tue and T 1,3 is false. As in the case of Example 5.1, it follows that each of the classes D 2,1 and D 2,2 split modulo 20 into fou teminal classes, with constant ν 5 (S(n, 4)) = 1, and one banching (nonconstant) class. This banching class will moeove split modulo 100 into fou teminal classes with constant ν 5 (S(n, 4)) = 2 and one banching class, and so on, at each m-level. The class D 1,3 coesponding to the vetex j = 3 equies futhe testing. The class {n N : n 3 mod 4} at the second level splits into five classes modulo 20: j = 3, 7, 11, 15, 19. Since (5.11) ( 1) 3 0 mod 5 2, then Poposition 4.5 implies that 5 2 divides S(D 2,j, k) fo j = 3, 7, 11, 15, 19. Hence, fo each of these, the following test is pefomed: fist, is S(j + 20t, 4) 0 mod 5 3? If not, then the vetex j is teminal, with ν 5 (S(n, 4)) = 2 fo evey n j mod 20. Then, does T 2,j hold? The only j fo which S(j+20t, 4) 0 mod 5 3 ae 3 and 11, so it is only necessay to test T 2,3 and T 2,11. Recall that β 2, is defined by (5.12) 20 1 + 5 2 β 1, + 5 3 β 2, mod 5 4 so β 2,, 1 4, is detemined by the following calculations: Thus we have 1 1 + 5 2 0 + 5 3 0 mod 5 4 2 20 1 + 5 2 3 + 5 3 3 mod 5 4 3 20 1 + 5 2 1 + 5 3 0 mod 5 4 4 20 1 + 5 2 1 + 5 3 2 mod 5 4 β 2,1 = 0, β 2,2 = 3, β 2,3 = 0, β 2,4 = 2.

18 A. BERRIZBEITIA ET AL. Then (5.13) 2 3 (3 + 3 5) 2 implies T 2,3 tue, and (5.14) 2 11 (3 + 3 5) 2 3 3 + 3 3 11 + 3 4 3 (1 + 2 5) = 1460 0 mod 5 2 4 4 11 (1 + 2 5) = 45649940 0 mod 5 2 4 implies T 2,11 tue. So pe Theoem 4.6, each of the classes j = 3, 11 at the second level splits modulo 100 into fou teminal classes with ν 5 (S(n, 4)) = 3 and one banching class. This banching class will moeove split modulo 500 into fou teminal classes with constant ν 5 (S(n, 4)) = 4 and one banching class, and so on, at each m-level. This completes ou chaacteization of this tee. Example 5.3. The last example descibed hee is p = 11 and k = 4, see Figue 13. 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 Figue 13. The 11-adic tee of S(n, 4) As in the pevious two examples, the fist level of this tee is fomed by the classes D 1,j, j = 0, 1, 2,, 9. The classes D 1,i, i = 0, 4, 5, 6, 7, 8, 9 ae teminal with valuation ν 5 (D 1,i ) = 0. The classes D 1,j fo each 1 j 3 equie test. Using Definition 3.3 and Example 5.2, the eade can veify that in this case we have β 1,1 = 0, β 1,2 = 5, β 1,3 = 0, and β 1,4 = 10. This infomation yields T 1,1 tue, T 1,2 tue, and T 1,3 false. Fom hee we gathe that each of the vetices j = 1, 2 splits modulo 110 into ten teminal classes, with constant ν 11 (S(n, 4)) = 1, and one banching (nonconstant) class. This banching class will moeove split modulo 1210 into ten teminal classes with constant ν 11 (S(n, 4)) = 2 and one banching class, and so on, at each m-level. Since T 1,3 is false, then the vetex j = 3 equies futhe testing. We poceed as in Example 5.2. At the second level, this vetex splits into eleven classes modulo 110. These ae j = 3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103.

p-adic VALUATION 19 Note that (5.15) 4 ( 1) 3 0 mod 11 2, theefoe by Poposition 4.5 we know that 11 2 divides S(D 2,j, k) fo each of these j. The following test is pefomed at each vetex: fist, is S(j +110t, 4) 0 mod 11 3? If not, then the vetex j is teminal, with ν 11 (S(n, 4)) = 2 fo evey n j mod 110. Then, does T 2,j hold? The only j fo which S(j + 110t, 4) 0 mod 11 3 is j = 3, so it is only necessay to test T 2,3. Recall that β 2, is defined by (5.16) 110 1 + 11 2 β 1, + 11 3 β 2, mod 11 4. A diect calculation gives Then 2 3 (5 + 1 11) 2 β 2,1 = 0, β 2,2 = 1, β 2,3 = 4, β 2,4 = 2. 3 3 (0 + 4 11) + 3 4 3 (10 + 2 11) = 1936 0 mod 11 2 4 and so T 2,3 is false. This implies that futhe testing is equied and so we still need to go anothe level down. The class {n N : n 3 mod 110} splits into eleven classes modulo 1210: Since (5.17) j = 3, 113, 223, 333, 443, 553, 663, 773, 883,993, 1103. ( 1) 3 0 mod 11 4, then 11 4 divides S(D 3,j ) fo each of these j. Once again, fo each of these classes, the following two test ae pefomed: fist, is S(j + 13310t, 4) 0 mod 11 5? If not, then the vetex j is teminal, with ν 11 (S(n, 4)) = 4 fo evey n j mod 1210. Then, does T 3,j hold? The only j fo which S(j +1210t, 4) 0 mod 11 5 ae 3 and 113. The calculation (5.18) β 3,1 = 0, β 3,2 = 3, β 3,3 = 0, β 3,4 = 6 implies T 3,3 and T 3,113 ae tue. So by Theoem 4.6, each of the classes j = 3, 113 splits modulo 13310 into ten teminal classes with ν(s(n, 4)) = 5 and one banching class, and so on, though m-levels. This completes ou chaacteization of this tee. This pocess is a systematic and vey easily automated way to summaize the stuctue of the p-adic valuation of S(n, k) fo fixed k < p. We conclude with an extension of Conjectue 1.1 fo any odd pime. Conjectue 5.4. Fix the index k. Given a pime p, its is conjectued that a) thee exists a level m 0,p (k) and an intege µ p (k) such that fo any m m 0,p (k), the numbe of non-constant classes at the level m is µ p (k), independently of m; b) moeove, fo each m m 0,p (k), each of the µ p (k) nonconstant classes splits into p 1 constant and one nonconstant subclass. The latte geneates the next level set.

20 A. BERRIZBEITIA ET AL. Suppoting softwae. This aticle is accompanied by the Mathematica package StilingTees.m available fom the webpages: http://www.math.utges.edu/~lmedina/ and http://www.math.tulane.edu/~vhm/ Acknowledgements. This poject stated at MSRI-UP duing the summe of 2008. The wok was suppoted by National Secuity Agency (NSA) gant H98230-08-1-0063, the National Science Foundation (NSF) gant 0754872, a donation fom the Gauss Reseach Foundation in Pueto Rico, and The Mathematical Sciences Reseach Institute (MSRI). The fouth autho acknowledges the patial suppot of NSF-DMS 0713836. Refeences [1] T. Amdebehan, D. Manna, and V. Moll. The 2-adic valuation of Stiling numbes. Expeimental Mathematics, 17:69 82, 2008. [2] H. W. Becke and J. Riodan. The aithmetic of Bell and Stiling numbes. Ame. J. Math., 70:385 394, 1948. [3] L. Calitz. Conguences fo genealized Bell and Stiling numbes. Duke Math. J., 22:193 205, 1955. [4] Y. H. H. Kwong. Minimum peiods of S(n, k) modulo M. The Fibonacci Quately, 27(3):217 221, 1989. [5] A. Nijenhuis and H. S. Wilf. Peiodicities of patition functions and Stiling numbes. J. Numbe Theoy, 25:308 312, 1987. Univesity of Wisconsin, Madison, WI 53706-1388 E-mail addess: beizbe@math.wisc.edu Depatment of Mathematics, Rutges Univesity, Piscataway, NJ 08854 and Depatment of Mathematics, Univesity of Pueto Rico, San Juan, PR 00931 E-mail addess: lmedina@math.utges.edu Columbia Univesity, New Yok, NY 10027 E-mail addess: acm2141@columbia.edu Depatment of Mathematics, Tulane Univesity, New Oleans, LA 70118 E-mail addess: vhm@math.tulane.edu Ohio State Univesity, Columbus, OH 43210 E-mail addess: noble@math.ohio-state.edu