Download formulas for the number of binomial coefficients divisible by a fixed

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
b + ap + ap2 +-h
(0 = a < p, 0 = b < p),
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<
Cxpkl+ ■■■+ cmPk"
(0 < ct < p,j = kx,j < ki+1 - kf).
r),
Received by the editors August 2, 1971.
AMS (A/05) subjectclassifications(1970).Primary 05A10,05A15.
Key words and phrases. Binomial coefficient, prime number.
© American Mathematical Society 1973
358
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359
NUMBER OF DIVISIBLE BINOMIAL COEFFICIENTS
We shall use the following rule, which was proved by Kummer [2,
p. 70]. Put
(1.2)
(1.3)
s = a0 + axp + --- + arpr
(0<ai<p),
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
1;
ak = ck+\,■'/c+2 = o,
— ck+xi ' ' ' >p
' ;
-1.
>c*
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)
rita+i)
Av =
l(ck + l)(ck+x + l)(c*+2 + 1)].
In the second case, we can take
ak = ck + \,---
,p -
am — cm + 1 ,--•
""jfc+l=
I;
,p — 1 ;
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
(2.2)
Bk_m =
Iífe+1)
[(ck + l)(ck+1 + l)(cm + l)(cm+l + 1)].
Thus we have
r-2
ö2(n) = ^(P(2-3)
k'° r_x
+ 2
m=t+2
ck-
ï)(p
-
ck+1)ck+2Ak
r-Z
2 (P - C*- l)Ck+l(P- Cm- 1) rn+X"k.mi
k=0
where Ak and Bkm are defined by (2.1) and (2.2) respectively.
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a's
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 -bl)(p - c)d
+ (p-al)b(p -cl)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,
(2) sk_}=ek_j+1=- ■-=ek_x=l,
(3) £*-m=£*-m+i=-
• •=6fc-i=l>
all other £'s=0;
all other £'s=0;
£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;
aT = 0, • - •, b — 1;
a¿ = 0, ••-,/>-1
ak = 0, • • •, a,
so there are (p—l)p'~1(a+l)b
other two cases, we have
%A\af + bpr) = (p-
(3.1)
(r -/
+ 1 < i < r - 1);
choices. Using similar reasoning in the
iy-»(a + l)b
+ (p - îy-^iè
+ 1) + (j - l)(p - \Yf-2ab
(k^j,r>k+j).
Similarly we have
Oj(apk + bpT)
= (p-
(3.3)
ljp*-*(fl + \)b + k(p - \)2pi-2ab
= (p - a - ly-^
= (p-
îy-Mè
+ k(p - l)2p>-2ab
+ l) + (p-
î)/»'-1^
= (p-
(3.7)
n = clPkl + c2pk> + ■■■ + cmpk™
= k +j),
j),
îy-1*
(k^j,r
l)P'-2(p - d)b + (r -j)(p
('}
We next evaluate 0¡(ri) for
(k<j,r
(k^j,r<k+
+ i) + (P-a-
+ (j - !)(/> - \)2r2ab
+j),
Dpj-2(p - a)b
+ (r - k - l)(p - l)2f-2ab
= (p-
(k<j,r>k
+ k =j),
- \)2p'-2ab
(k<j,r<k+j,r^j).
(kx £/,
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ki+1 -
kt >j).
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
= 0, ■••,/>-
1
= 0, • • • , cv - 1
(» » i"w— f„, 1 < w < u),
(v = ia-h,l^h<tw-
1,1 ^w^u),
(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,
(3.10)
then (3.8) becomes
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/
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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-
If ^=0,
%
- 1)V-"£U.
ki+1—kj>j, then
(3.13)
ex«)= £iI (;
~ ¡W - DV-BG».
\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-
(3-14)
+
l)a(p - a - l)pj-\a
then (3.11) becomes
+ l)m~2
S (j¡" J) (")(P - l)V-(a + l)m-Ma";
(3.12) becomes
(3.15)
6fin) = (p- W'^a
+ I)™"1
+ (m - l)(p -al)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), 299-320. MR 40 #2554.
2. L. E. Dickson, History of the theory of numbers. Vol. 1, Publication no. 256,
Carnegie Institution of Washington, Washington, D.C., 1919.
3. N. J. Fine, Binomial coefficientsmodulo aprime, Amer. Math. Monthly 54 (1947),
589-592. MR 9, 331.
4. F. T. Howard,
The number of binomial coefficients divisible by a fixed power of 2,
Proc. Amer. Math. Soc. 29 (1971),236-242.
Department of Mathematics, Wake Forest University, Winston-Salem, North
Carolina 27109
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