FORMULAS FOR THE NUMBER OF BINOMIAL COEFFICIENTS DIVISIBLE BY A FIXED POWER OF A PRIME

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 37, Number 2, February 1973 FORMULAS FOR THE NUMBER OF BINOMIAL COEFFICIENTS DIVISIBLE BY A FIXED POWER OF A PRIME F. T. HOWARD Abstract. Define 0,(n) as the number of binomial coefficients (?) divisible by exactly/»'. A formula for 02(n) is found, for all n, and formulas for 6,(n) for n=apk+bpr and n=c,pki+- +cmpkm (&i=./'» +i~ ísí/'for /=1,, m 1) are derived. 1. Introduction. Let p be a fixed prime and let d}(ri) denote the number of binomial coefficients (") (s=0, 1,, ri) divisible by exactly p'. If we put (1.1) n = c0 + clp + + ctpr (0 = c, < p) it is well known [3] that W = (Co + \)(ci + 1) (cr + 1). The evaluation of &}(ri) for arbitrary y appears to be more difficult, however. Carlitz [1] has proved that r-l fli(«) = 2 (co + O " ' (ck-i + \)(p -ck- \)ck+x(ck+2 + 1) (cr + 1) *=o and he has found formulas for 03(«) for the following values of«: apr + bpr+1 (0 = a < p, 0 = b < p), b + ap + ap2 +-h apt+i (0 < a < p, b = a or a - 1). The writer [4] has considered this problem for p=2 and has found formulas for 0,-(/i), l^y"^4, and for arbitrary y has evaluated 6,-(/i) for a number of special values of n. These formulas are valid only for p=2, however. In this paper we find formulas for d2(n) for all n and for 03(n) for the following values of n : apk + bpt (0<a<p,0<b<p,k< r), Cxpkl + + cmpk" (0 < ct < p,j = kx,j < ki+1 - kf). Received by the editors August 2, AMS (A/05) subject classifications (1970). Primary 05A10, 05A15. Key words and phrases. Binomial coefficient, prime number. 358 American Mathematical Society 1973

2 NUMBER OF DIVISIBLE BINOMIAL COEFFICIENTS 359 We shall use the following rule, which was proved by Kummer [2, p. 70]. Put (1.2) s = a0 + axp arpr (0<ai<p), (1.3) n - s = b0 + bxp + + btpr (0 ^ bt < p), a0 + b0 c0 + e0p, e0 + ax + bx = Cx +,p,, r_r + ar + bt= ct + srp, where each et=0 or 1. Let N he the exponent of the highest power of p that divides ("). Then we have N=e0+Sx+- -+er. 2. Evaluation of 62(n). If n is given by (1.1) and s and n s are given by (1.2) and (1.3), it is clear that A^=2 if and only if exactly two of the e's are equal to 1, and r=0. There are two possibilities. Either ek= ek+1= 1 for some 0^k^r 2 and all other e's=0 or ek= 1, em= 1 for some 0_&^r 3, &+2_m_r 1 and all other e's=0. In the first case we can take ak = ck+\,- 1; ck+xi ' ' ' >p ' ; '/c+2 = o, >c* -1. So we have (p ck l)(p ck+1)ck+2 choices and the remaining a's can be chosen in Ak ways, where (2.1) Av = rita+i) l(ck + l)(ck+x + l)(c*+2 + 1)]. In the second case, we can take ak = ck + \,---,p - I; am cm + 1,--,p 1 ; ""jfc+l = 0, -1, = 0,,cm+1-1, so there are (p ck l)ck+1(p cm l)cm+1 choices. The remaining can be selected in Bk m ways, where a's (2.2) Bk_m = Iífe+1) [(ck + l)(ck+1 + l)(cm + l)(cm+l + 1)]. Thus we have r-2 ö2(n) = ^(P- ck- ï)(p - ck+1)ck+2ak (2-3) k' r_x r-z (P - C* - l)ck+l(p - Cm - 1) m=t+2 k=0 where Ak and Bkm are defined by (2.1) and (2.2) respectively. rn+x"k.mi

3 360 F. T. HOWARD [February For example, 62(a + bp + cp2) = (p-a- l)(p - b)c, 02(a + bp + cp2 + dp3) = (p-a- \)(p - b)c(d + 1) + (a + \)(p -b- l)(p - c)d + (p-a- l)b(p -c- l)d. This method does not appear to be very practical for evaluating 6 (ri) íoxj>2. 3. Special evaluations. We can use Kummer's theorem to evaluate Qj(apk+bpT), where r>k, 0<a<p, 0<b<p. Suppose fc>y and r k>j. Then there are three ways to have exactly y of the f's equal to 1 : (1) r_3=er_i+i=- -=er_i=l, all other 's=0; (2) sk_}=ek_j+1=- -=ek_x=l, all other 's=0; (3) *-m= *-m+i=- =6fc-i=l> r-a=- -=^-1=1. \ú új-l, h= j m, all other e's=0. If j is given by (1.2), in the first case we can take ar_i = 1,,/>- 1; a = 0, -,/>-1 (r -/ + 1 < i < r - 1); at = 0, -, b 1 ; ak = 0,, a, so there are (p l)p'~1(a+l)b other two cases, we have choices. Using similar reasoning in the %A\af + bpr) = (p- iy-»(a + l)b (3.1) + (p - îy-^iè + 1) + (j - l)(p - \Yf-2ab (k^j,r>k+j). Similarly we have Oj(apk + bpt) = (p- ljp*-*(fl + \)b + k(p - \)2pi-2ab (k<j,r>k +j), (3.3) = (p - a - ly-^ + k(p - l)2p>-2ab (k<j,r = k +j), = (p- îy-mè + l) + (p- Dpj-2(p - a)b + (r - k - l)(p - l)2f-2ab (k^j,r<k+ j), = (p- î)/»'-1^ + i) + (P-a- îy-1* + (j -!)(/> - \)2r2ab (k^j,r + k =j), = (p- l)p'-2(p - d)b + (r -j)(p - \)2p'-2ab ('} We next evaluate 0 (ri) for (k<j,r<k+j,r^j). (3.7) n = clpkl + c2pk> + + cmpk (kx /, ki+1 - kt >j).

4 1973] NUMBER OF DIVISIBLE BINOMIAL COEFFICIENTS 361 Using Kummer's theorem, we need to determine the number of ways we can have exactly y of the s's equal to 1. Let 1 ^u^m and choose u of the c/s. Call them cfl,, ciu. Assign to each c^ a number tw, 1 5j tw, such that fi+i2+' ' ' + iu=/ This can be done in ( I*) ways, since there are ( -Î) different ways of distributingy nondistinct objects into u distinct cells with no cell left empty. We wish to have =1 (v=iw h, l<h^tw, 1^w_m) and all other e's equal to 0. If s is given by (1.2) we can take a = 1,,p - 1 (»» i"w f, 1 < w < u), = 0,,/>- 1 (v = ia-h,l^h<tw- 1,1 ^w^u), = 0,, cv - 1 (p - /, 1 w g «). Thus for a given «and a given selection ix,, i, there are {I Z\)(p- l)y-"^ <*.(* + o (cm + i)kch + o (cfii +1) different ways to have y of the 's equal to 1. Therefore (3.8) ein) = 2 (; ~ ta - i)v~u» u^x \«- 1/ where n is given by (3.7) and (3.9) Eu = J cfl cjq + 1) (cm + l)/[(c,, + 1) (ciu + 1)], the sum being over all subsets {ilt, i } of {1,, m) such that í'i<'2<-- <' For example, if n=apkl+bpk*+cpk*, kx^j, k2 kx>j, k3 k2>j, then 0,(10 = (p - \)pj-x[a(b + \)(c + 1) + (a + l)6(c + 1) + (a+l)(6+l)c] + 0' - l)(/>-l)y-2[a >(c + 1) + a(è + \)c +(a+ \)bc] + (J~l)(p-l)3pi-3abc. If n is given by (3.7) and Ci = c2=- cr^sa, then (3.8) becomes (3.10) din) = 2 ( I J) (")(P - 1)"F^(«+ Vm~UaU- If n is given by (3.7), except that rv,=y 1 and ki+1 ki=j, we have, by an argument similar to the one above, (3.11) din) = pj-xf +2(j~ Î)(P - 1)V~", uf2 \u - 1/

5 362 F. T. HOWARD where Eu is defined by (3.9) and m l F=2(P- i- Ifcwfci + l)"-(c. + l)/[fo + IXcm + 1)]- If kx>:j and ki+1 k~j, then 0/n) = (p - l)/>i-1c1(c2 + 1) (cm + 1) (3-12) + p«f + Z(J- % - 1)V-" U. If ^=0, ki+1 kj>j, then (3.13) ex«) = I (; ~ W - DV-BG». i \u - 1/ where Gu = 2 c ciu(cx + 1) (cm + l)/[(c;, + 1) (c, + 1)], the sum being over all subsets {ilt, iu} of {2,, m} such that *1<*2<' - <'«If«is given by (3.7) and Cx=c2=- -=cm=a, 8fin) = (m- l)a(p - a - l)pj-\a + l)m~2 (3-14) + (3.12) becomes then (3.11) becomes S (j " J) (")(P - l)v-(a + l)m-ma"; 6fin) = (p- W'^a + I) "1 (3.15) + (m - l)(p -a- l)a(a + 1) Y"1 +? («-1) (r)(p "i)v_u(a+i)m_ua"; (3.13) becomes (3.16) 0/n) = 2 (^ Z\)(m~ l)ip - DV-'C* + l)m~"«"- References 1. L. Carlitz, The number of binomial coefficients divisible by a fixed power of aprime, Rend. Circ. Mat. Palermo (2) 16 (1967), MR 40 # L. E. Dickson, History of the theory of numbers. Vol. 1, Publication no. 256, Carnegie Institution of Washington, Washington, D.C., N. J. Fine, Binomial coefficients modulo aprime, Amer. Math. Monthly 54 (1947), MR 9, F. T. Howard, The number of binomial coefficients divisible by a fixed power of 2, Proc. Amer. Math. Soc. 29 (1971), Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109