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IDENTITIES RELATING THE NUMBER OF PARTITIONS INTO AN EVEN AND ODD NUMBER OF PARTS, II DEAW R. HICKERSON University of California, Davis, California 95616 Definition. If / > 0 and n > 1, let q?(n) be the number of partitions of/? into an even number of parts, where each part occurs at most/times. Let q°(n) be the number of partitions of A? into an odd numberof parts, where each part occurs at most/times. If/ > 0, let^f(O) = 1 and ^ ( 0 ) = 0. Definition. If / > 0 and n > 0, let A{(n) = qf(n) - qf(n). The purpose of this paper is to determine Ai(n) when / is any odd positive integer. The only cases previously known were / = 1, proved by Euler (see [1]), / = 3, proved by this writer (see [2]), and/= 5 and 7, proved by Alder and Muwafi (see [3]). Definition. If s, t, u are positive integers with s odd and 1 < s < t, and n is an integer, let fs> t}U (n) be the number of partitions of n in which each odd part occurs at most once and is ^ ±s (mod It) and in which each even part is divisible by It and occurs < u times. Theorem. If s, t, u are positive integers with s odd and 1 < s < t, and n is an integer, then &2tu-i(n) = (~VnZ fs,t,u(n-tj2-(t-s)j). Proof. z A2tu~i(n)xn = n lzj^1 = n n-*1)- n r / - * ^ w ^ v . . ^ - ^ 2\j = JJ (1 2t\j _ x2tj+s)(1 _ x2tj+2t-s)(1 _ x2tj+2t)mYY(J_xJh j i j>0 J j>l 2t\j where the last equality follows from Jacobi's identity with k = tant\z = t-s. tj + (t-s)j Hence, when we substitute -x forx, we obtain , iia+xj+x2j+-+x(u~1)j), d-xh- JJ j>l 2\) 2\) _ j£±s (mod 2t) 2 j j>l j>l 2kj 2t\j j^+s (mod 2t) = E (-Djxtil+(t-s)^ n j Y\(l+xJ+x2j+...+x(u-l)j) =/ (mod 2). Since s is odd, IDENTITIES RELATING THE NUMBER OF PARTITIONS (-1)nA2tu„l(n)xn £ =£ x^ n + ^ - EI j d + x')- El j (1 + FEB. 1978 x3+x2i+..-+x(»-l)i) j j>l 2\j f£±s (mod 2t) j>l 2t\j = 2 > ^ ^ ' . £ fs,t,Jm)xm, j m from which the theorem follows immediately. Corollary 1. \is and t are positive integers with s odd and 1 < s < t, and n is an integer, then *2t-i(n) = (~1)n E fs,t,l(n-tf2-(t-s)/). j Note that fS)t,i(n) is the number of partitions of n into distinct odd parts ^ ± s (mod It), Proof. Ls\u= 1 in the theorem. Letting^ = 1 and t = 3 yields Theorem 1 of [ 3 ] . 2. If i> 2 and/? is an integer, then (-J)nAi(n)> Corollary 0. Proof. For even i, this follows from Theorem 3 of [ 2 ] ; for odd i, it follows by letting s = 1 and t = (i + D/2 in Corollary 1. Corollary 3. If s and t are positive integers with s odd and 1 <s <t, and/? is an integer, then A4t-lM = <-Dn £ j fs,t,2(n-tj2-(t-s)j). Note that fSyt,2(n) ' s t n e number of partitions of/? into distinct parts which are either odd but ^±s (mod 2f) or which are divisible by It Proof Corollary Let £/ = 2 in the theorem. 4. If u is a positive integer and n is an integer, then ±4u-lM = (-Dn L h,2,Jn-2j2-j). j Note that fi}2,u(n) Proof. is the number of partitions of/7 into parts divisible by 4, where each part occurs <u times. Lets= 7, f = ^ in the theorem. Letting u = 1 yields Theorem 2 of [2] and u = 2, Theorem 2 of [ 3 ] . REFERENCES 1. Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 3rd ed., pp. 221-222, Wiley, New York, 1972. 2. Dean R. Hickerson, "Identities Relating the Number of Partitions into an Even and Odd Number of Parts," J. Combinatorial Theory, Section A (1973), pp. 351-353. 3. Henry L. Alder and Amin A. Muwafi, "Identities Relating the Number of Partitions into an Even and Odd Number of Parts," The Fibonacci Quarterly, Vol. 13, No. 2 (1975), pp. 337-339.