SUMMATION FORMULAE FOR
MULTINOMIAL COEFFICIENTS
SELMO TAUBER
Portland State College, Portland, Oregon
1.
INTRODUCTION
In [ 1] we have given some h i s t o r i c a l background to the m u l t i n o m i a l
coefficients and proved some of the b a s i c s u m m a t i o n f o r m u l a e .
More
of the s u m m a t i o n formulae can be found in [ 2 ] . In this p a p e r we shall
prove additional r e l a t i o n s involving m u l t i n o m i a l coefficients.
Some of
t h e s e can be c o n s i d e r e d a s g e n e r a l i z a t i o n s of c o r r e s p o n d i n g f o r m u l a e
for binomial coefficients.
2.
We shall r e f e r to [3] for t h e s e f o r m u l a e .
FIRST SET OF FORMULAE
In o r d e r to simplify the notation used in [l] , at least for the
proof, we shall w r i t e
n
n
N:/n
ks:
= (
and, for,
(
k
k
V
8=1
k1+k-+.-..+k
k
k
N
Z
= N+l,
\ - l
),
k
' " "
with,
n
s
kg = N ,
8=1
we shall have the simplified notation
k
k ) = [N.(k.-l),k
]
.
Under t h e s e conditions equation (6) of [ l ] can be w r i t t e n
n
(1)
2
For
[N,(k.-l),kn]
=
[N+l,kn]
.
j=l
0 = p = n, we can w r i t e (1) in the form
p-1
2[N.(krl).kpfkn]+[N,kp-lfkn]+
j=l
n
X
[N,kp5(krl),kn]
j=p+l
= [~N+1, k , k 1
L
p n J
and s i m i l a r r e l a t i o n s for
.
N - l , N-2,
aso
, N-q, . . . , N-k , thus,
P
[LN - l , (k.-l),k
- l , k n Jl + [LN - l , k p -2,k n J1+ S fLN - l , k p - l , ( k .j - l ) , k n1J
j
' p
j=l
r
,
, ,
-i
J=P+1
95
96
SUMMATION FORMULAE FOR
p-1
1 [N-q,(kj-l)Ap-q,kjH{N-q,kp-q-l,kn]+
j=l
n
1
[N-q, k p - q , ( k . - l ) , k j
j=p+l
= [N-q+l,k
p-1
-q.kj
n
2 [ N - k , (k.-l), 0,kn]+
j=l
2
J=P
By adding the f i r s t
q
April
[N-.k , 0 , ( k - l ) , k n ] = [ N - k + l , 0 , k n ]
+1
q equations and simplifying we obtain
"p-1
n
2
2 [NTQ, ( k . - l ) , k -a, k ] + 2 [N-a, k - a , ( k . - l ) , k 1
a=0 J = l
J=P+1
o r , using the c l a s s i c a l notation,
q rp-1
(2)
N-a
I ( K 1 , K 0 , 9 . . , K . . . , K . - l , l c . - , # . . , K - a, . . . , K, +
1 L
j-i j
j+1
p
n
2
a=0|
x
.=
(
+1
-
k
N a
"
k
l' 2'
kp-a,
)
....k.^.k.-l.k..^....,^
N+l
N-q
( k, , k-j, o o » , k , • . . , k
k, , k~, • . • c, k- - q , . . • , k
V
P
F o r q = k , we obtain
P
(3)
rp-i
2 2 (k , ,
a=0
u=i
i
N-a
K~, • • . , K. , j K. - 1 , k. ,, , • • • , k - a, . . . , k )
c*
j -1
j
J"'"-'-
n
. 2 . ( k-,
P
N-a
9
iC0, . • . , k
- CL, » „ . , K .
1 Z
p
j= P +l
= ( R., , K^> # . . ,N+l
K- f •
I L
p
1
• . , K
n
.
, K . - 1 , K .
j-1
+
"
j
1
, . . . , K
j+1
n
.
1965
MULTINOMIAL COEFFICIENTS
97
It "will be noted that in both (2) and. (3) the sum i s independent of p, thus
by s u m m i n g on p we obtain
k _
n
p -p-i
(4)
2
2
p=l
s ( k., , k~,
i
a=0|j = l
n
L
N-a
• . . , k. 1 , k . - l , k . . , , . . . , k - a j e
j-l j
j+1
p
+
N-a
2
(l k, , k ? , . . . , k - a , . . . , k. , , k . - l , k. , , . . . , k
i+r
J-l' J
j=p+l
N+l
= n( k , k , . . . , k , . . . , k
x
2
g
n
For
n = 2, (2) and (3) r e d u c e to (3) and (4) of [3] , p. 246.
3.
SECOND SET OF FORMULAE
Consider the f o r m u l a e leading to (2) and (3). If we multiply the
f i r s t r e l a t i o n by +1, the second by - 1 , . . . , the ( q + l ) - t h
(-1) , e t c . , . . .
2
we obtain
p-1
q
(5)
r e l a t i o n by
(-1)
N-a
2 (
)
+
k j , ^ , .B.,k.^rk.-l,k.+r ...,kp-a, ...,kn
a=0L
n
,=
N-a
+1
ki'V-'-'V'0"
"•,kJ-l,kj"1,kJ+l""",kn
q
N-l
2 2 ( - l ) a ( k, , k . . .N-a+l
, k - a , . . . , k )+d
^ , k-, . . . , k , . . . , k ) +
oJ
1 2
p
n l Z
p
n
a=l
(-l)q+1(
and,
k
(6)
N
k-. » k_ j . o . , k
P
2
a=0
r
*
"q
)
-q- Ij . . . , k
D-l
* \.k
2
(
j= P +l
k
2
kj.^kM'Vl
a , , , , k
)
n
+
N-a
k r k2, . . - » k p - a » • • • ' k j - l , k j " 1 , k j + l ' ' ' * , k n
P
N+l
N-a+l
22 (
i k, , k~, . . . , k - a, . . o , k
a=l
V
1
2
p
n
k, , k 9 , . . - . , k , • . . , k )
I
2
p
n
.
April
SUMMATION FORMULAE FOR
98
(5) and (6) for n = 2 r e d u c e to (7) of [3] , p . 247.
4.
THIRD SET OF FORMULAE
Using the notation of section 2 we w r i t e for
n
2 k = N+l, k = 0 ,
^
s
p
s=l
p-1
2 [N, ( k . - i ) , o , k n ]
n
+
2
[N, 0, ( k . - l ) , k n ] = [ N + l , 0 , k
]
J=P+1
2 [N+l, (k.-l), l , k n ] + [ N + l , 0 , k n ] +
2
[N+l, l . ( k . - l ) , k n ]= [N+2, l . k j
j=p+l
j=
2 [N+q, (k.-l), q, k n ]+ [N+q, q - l , k j +
2 [N+q, q, (k.-l), k j = [N+q+1, q, k n ]
J=P+1
j=
P"
2 [N+h, (k.-l).h, k ]+[N+h, h - l , k n ] +
2
[N+h, h, (k.-l), k 1= [N+h+1, h, k n ] ,
j=p+l
j=
By adding and simplifying we obtain
q
2
p-1
2 [N+a, ( k . - l ) , a, k ] +
2
[N+a, a, ( k . - l ) , k
[N+q,q,k n ] ,
]
J=l
j=P+l
or, using the c l a s s i c a l notation
p-1
Q=0
(7)
2
S
A
(
• i
n
2
+
k , , k 0 , . . . , k . ^ k . - ^ k . , . , . . . , a, . . . , k '
1
(k
j=p+i
2
j-1
l ' k2
a
j
l+l
k
N+a
j - l * k j - 1 ' k j+l
N+q+1
k
k
k
l' 2"-" p-l'
q + 1
'kP+l',','kn
n
.
k
n
1965
MULTINOMIAL COEFFICIENTS
99
and s i m i l a r l y
h -p-1
(8)
N+a
1
•K-i > K _ , . , . , K .
1 , K . - I , K ,
i . ,
1
L
l-l
i
l+l
a=q|_j=l
J
J
J
n
i
1 '
j=p+i
• • • » a»
2'
• ° " '
,
l'
k
2'•'•'kp~1'
(
k
For
.
0
, a ,
B
. . , K
n
N+a
.(
k
0
h+1
i_i
' k.~l,
n+h+1
' kp+l'
i-i-i » • " • »
-r,
,
B
• • ' kn
n+q-1
rk2' •••,kp-i,q'1'kP+r
^
e e , , k
n
n = 2 (8) r e d u c e s to (11) of [3] p. 248.
5.
FOURTH SET OF FORMULAE
(8) of [1] can be simplified in writing by introducing the notation
k ,
k
1
2
^n-1
n-1
s
1
(9)
...
1
1
= (n
•
2 )
j =0 j =0
j . n
.s=l
j =0
J
J
J
J
l
2
n-1=0
s
II o p e r a t e s on the o p e r a t o r 2 . Under t h e s e conditions (8) of
where
[ l ] can be w r i t t e n for,
n
n
1
k g = p+q,
s=l
i
j
g
=p
,
s=l
k
n-1
s
(io) ( n • 2
s=l
1
j =0
)(,h'h'-'->
•
Let us substitute
p+r
for
)-d.
= k
j )(,.
k
n l "V V j 2 ' • • • ' V j n
p in (10),
we obtain for
p+q
l ' k 2' • • • ' k n
(j. , j _ , . . . , j )
n-1
p+r
'h'h'""3n
with
) = ( n • 2 )L
\ t=l
"
h
j
h
h
h, =0 ' J r l ' 2 - 2 ' - - " V n
)(vh
l'h2'-'-'hr
100
SUMMATION FORMULAE FOR
2
h. = r ,
2
i=l
j . = p+r,
2
i=l
April
k. = p + q + r
,
i=l
s o t h a t s u b s t i t u t i n g i n t o (10) w e o b t a i n
n-1
(ii)
s
( n
•
s=l
S
)(n
j =0
J
• v(
J
n-1
t
• 2
t=l
)(, _ k
K
Jl
!
h =0
:
i »
!
p
k
JT
.
•••JJ
J
n
S J
2
k
2
)(k
K
n
i
q
. ) .
k
K, - J , , . . . , K -J
1 Jl
t
s
r
) = l/
P+cl+r
)
h,,h9, .. .,h ;
kl3k0, . . . ,k '
1 Z
n
1 Z
n
'
M o r e g e n e r a l l y a s c a n be p r o v e d by i n d u c t i o n w e c a n w r i t e
m-1
(12)
n
i=l
J
k
n-1
'
( n
2
'
i=l
i, j
k
m-1
}
n
=0
k
i=l
in
m, 1
J j l
-k
k
J + 1 , 1
J
J
j+l,i
. /
(
x _ , ^1
m, 2
m, n
q
11
-kqj
'
J+1'2
2
2 * * '
12
q
k
J
'
n
k
J + 1
m v
In
where,
S
<kj,t~Vl,t)
= q
j'
f
°rj
J= 1 . 2 - - - n
t=l
REFERENCES
S. T a u b e r , " O n M u l t i n o m i a l C o e f f i c i e n t s , " A m e r , M a t h . MonthLy,
70(1963),
1058-1063,
L. C a r l i t z ,
" S u m s of P r o d u c t s
Elemente der Mathematik,
E. Netto,
Chelsea,
Lehrbuch der
of
Multinomial
18(1963), ppe
Combinatorik,
N . Y. , 19-58.
xxxxxxxxxxxxxxx
Coefficients, "
37-39.
r e p r i n t of s e c o n d Ed„ ,
>•
'n