JOURNAL OF COMBINATORIAL THEORY (A) 15, 351-353 (1973)

Note

Identities Relating the Number of Partitions

into an Even and Odd Number of Parts

DEAN R. HICKERSON

Department of Mathematics, University of California, Davis, California 9.5616

Communicated by the Managing Editors

Received April 18, 1972

DEFINITION. If i > 0 and n >, 1, let qi”(n) be the number of partitions of n into an even number of parts, where each part occurs at most i times. Let qf(n) be the number of partitions of n into an odd number of parts, where each part occurs at most i times. If i >, 0, let qie(0) = 1 and qio(o) = 0.

DEFINITION. If i > 0 and n > 0, let d&r) = qie(n) - q&z). The purpose of this paper is to determine d,(n), when i = 3 or an even number, the case i = 1 being a known result.

The generating function for di is given by

z. d,(n) xn = (1 - x + x2 - *** + (-l)i x”)

.(1-~2+~-...+(-l)ix2i)

. (1 - x3 + y3 _ . . . + (-l>i x3i) . . .

= fj(1 - xi + $39 - *** + (-l)ix”j)

THEOREM 1.

if n = (3j2 j-j)/2 for somej = 0, 1,2,..., otherwise.

This is a well-known theorem. See [l].

351 Copyright 0 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.

352 HICKERSON

THEOREM 2.

if 12 = (j” + j)/2for s0me.i = 0, 1,2 ,..., otherwise.

Proof. The generating function for A, is given by

m 1 -.x43' f’ A,(n)x” = n-I-- =

fi (1 - ~j)(l - x4j)

n=n j=l f x3 j=l 1 - $j

= fp - X4j+l)(l - -x4i+3)(1 - ,4i'4)}.

It is proved in [2, Theorem 3541 that

sg (1 - p)/fi (1 - -3-l) = f xiCj+l)/2e i=l i=O

Substituting -x for x and rearranging, we obtain

to (-l)j(ji-l)/2~j(i+l)/~ = fi (1 _ x4i) fi (1 _ x4i-2)/fi (1 + x2i-1)

i=l i=l i=l

= fi (1 -X4i) fi (1 - xzi-1)

i-1 i=l

= fyi - X4’+l)(1 - x4i+3)(1 _ X4i+4)},

Therefore,

f A,($ xn = -f (-l)~ci+1,/zx~ci+1,/2.

V&=0 j=o

By equating coefficients of xn on both sides of this equation, we obtain the theorem.

DEFINITION. If i > 0 is even, and n > 1, let p/(n) be the number of partitions of y1 into distinct odd parts which are not divisible by i + 1. Let pid(0) = 1.

THEOREM 3. rf i 3 0 is even and n > 0, then A&z) = (- 1)” pi”(n).

PARTITIONS INTO AN EVEN AND ODD NUMBER OF PARTS 353

Proof. The generating function for di is given by

pi(~) x” = fi l + x(d+.l’i = g (1 + xi)-’ I + x3

(i-tl).d

g ((1 + x9(1 + x”j>(l + x”j)(l + x”j) . ..}-1. 2fi

(itl)<i

Since

(1 + x9(1 + x”i)(l + x4j)(l + 9’) * *. = (1 - xy,

it follows that

Then

= 5 (1 - (-x)j) = I-I (1 + x9.

2ri 23 (i+l)ri (i+l)fj

The last product is clearly the generating function for pla, so that

i. (- 1)” 44 xn = i. PidW xn*

Therefore, (- 1)” d&z) = pi”(n) if n > 0, which completes the proof.

REFERENCES

1. IVAN N~VEN AND HERFIERT S. ZUCKERMAN, “An Introduction to the Theory of Num- bers,” 2nd ed., pp. 224-226, Wiley, New York, 1966.

2. G. H. HARDY AND E. M. WRIGHT, “An Introduction to the Theory of Numbers,” p. 282, Oxford University Press, London 1938.

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