DYSON'S CRANK OF A PARTITION

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1 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 8, Number, April 988 DYSON'S CRANK OF A PARTITION GEORGE E. ANDREWS AND F. G. GARVAN. Introduction. In [], F. J. Dyson defined the rank of a partition as the largest part minus the number of parts. He let 7V(ra, t, n) denote the number of partitions of n of rank congruent to m modulo, and he conjectured (.) 7V(m, 5,5n + ) = ±p(5n + ), < m < ; (.) JV(ra,7,7n + 5) = ±p(7n + 5), < m <, where p(n) is the total number of partitions of n [, Chapter ]. These conjectures were subsequently proved by Atkin and Swinnerton-Dyer []. Dyson [] went on to observe that the rank did not separate the partition of In -I- into equal classes even though Ramanujan's congruence (.) p(lln + )= (mod ) holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + ) in the same way that (.) and (.) treat the primes 5 and 7. In [, 5], one of us was able to find a crank relative to vector partitions as follows: For a partition 7r, let #(7r) be the number of parts of ir and cr{n) be the sum of the parts of ir (or the number ir is partitioning) with the convention #(</>) = <j{<t>) = for the empty partition, of. Let V = {(7Ti,7T,7T) 7Ti is a partition into distinct parts, 7T,7T are unrestricted partitions}. We shall call the elements of V vector partitions. For 7? = (7Ti, 7r, ^) in V we define the sum of parts, s, a weight, a;, and a crank, r, by (.) 8(*) = ör(tti) + a{7t ) + (T{TT ), (.5) <7r) = (-l)#^), (.) r(7f) = #(7r )-#(7r ). We say 7? is a vector partition of n if 5(7?) = n. For example, if #=(5+ +, + +, + + ) Received by the editors August, Mathematics Subject Classification (985 Revision). Primary P7. First author partially supported by National Science Foundation Grant DMS American Mathematical Society 7-979/88 $. + $.5 per page

2 8 G. E. ANDREWS AND F. G. GARVAN then 5(7?) = 9, a;(7?) =, r(ir) = and 7? is a vector partition of 9. The number of vector partitions of n (counted according to the weight u) with crank m is denoted by Nv{m, n), so that (.7) N v (m,n) = w(tf). fiev r(rt)=m The number of vector partitions of n (counted according to the weight u) with crank congruent to k modulo t is denoted by ivy(fc,,n), so that (.8) N v (k,t,n)= ^ N v {mt + k,n)= ^ u(jf). m=-oo jf V r(7f)=fc(modt) By considering the transformation that interchanges 7T and n^ we have (.9) Ny (m, n) = Ny (-m, n) so that (.) N v (t -ra, *, n) = N v (m, t, n). We have the following generating function for ivv(m,n): (.) Y Y N v {m,n)z m q n = TT K \* ) t,. m=-oon= n=l V ^ /V * ' By putting z = in (.) we find (.) ]T #v(m,n)=p(n). m= oo VECTOR-CRANK THEOREM (GARVAN [, 5]). (.) iv v (,5,5n + ) = JV v (l,5,5n + ) = -. = 7V v (,5,5n + ) = p ( 5n + )? (.) p(7n + 5) JVv(,7,7n + 5) = ivv(l,7,7n + 5) = --- = Av( > 7,7n + 5) = 7 p(lln + ) (.5) JV v (,, lin + ) = = iv v (,, lin -f ) = The above still leaves open the question of whether there is a crank for ordinary partitions. The answer is "yes" when the crank is denned as follows: DEFINITION. For a partition 7T, let l(ir) denote the largest part of 7r, U(IT) denote the number of ones in 7r, and ^(TT) denote the number of parts of w larger than ÜJ(TT). The crank c(w) is given by \ /i(ir) - u(ir) if W(TT) >.

3 Our main result is the following. DYSON'S CRANK OF A PARTITION 9 THEOREM. The number of partitions nofn with c(?r) = m is ivv(ra, n) for alln >. Obviously, in light of the Vector-Crank Theorem, we see that Theorem supplies the crank asked for by Dyson.. PROOF OF THEOREM. We shall require the standard notation of ^-series: (.) (A; g ) n = (^) n = n ( ( _"^) (= ( - A){ Aq) ( - Aq n ~ x ) when n is a positive integer), and (.) (A;g)oo = (A)oo = n(l-v)- = We now transform (.): E V^ M ( Wm» _ ( ~g) (^)oo m=-oon= (9 ;g) (q/z) c (-q) f, <7^ (*«)»,tï(9 ;9)i-i(V + )oo' As was noted in (.), when we set z = the series on the left of (.) reduces to the generating function for p{n). For j >, the jth term in the sum on the right is j times z- jq i+i+...+i ( - g)(l - «8)... (i _ 9i)(i _ ^i+i)(i _ ^+)..." The standard techniques of partition theory [, Chapter ] show that this expression generates partitions with uj(ir) = j and the exponent on z is clearly //(7r) - u;(7r), i.e. c(7r), since j >. Thus we must interpret (i-«) (l-^)(l-^ )( _^). as the generating function for partitions without ones. By considering conjugate partitions, we note that (l-zq)(l-zq*)(l-zq*)

4 7 G. E. ANDREWS AND F. G. GARVAN generates all partitions with the exponent on z counting the largest part, and for integers larger than Q (-*)(-* 9 )(-Zfl)... generates partitions with at least one appearing again with the exponent on z counting the largest part. Note that this interpretation fails for because this is the unique instance in which introducing a into the partitions of n alters the largest part. Hence (*tf)oo counts (for n > ) the number of partitions with no ones and with the exponent on z being the largest part of the partition l(ir) = C(T). Thus in the double series expansion of (l-q) fk g»w (*«)~,ti(«a ;«)i-i(*«, ' + )~ we see that the coefficient of z m q n (n > ) is the number of ordinary partitions of n in which c(7r) = m. Therefore by (.), we have Theorem. D. Conclusion. We can't resist exhibiting c(ir) for the first instance of (.). partitions of l(ir) w(7r) ^(TT) C(IT) As Theorem together with (.5) predicts, C(TT) provides eleven different residue classes modulo. REFERENCES. G. E. Andrews, The theory of partitions, Encyclopedia of Mathematics and Its Applications, Vol. (G.-C. Rota, éd.), Addison-Wesley, Reading, 97 (Reprinted: Cambridge Univ. Press, London and New York, 98).. A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. () (95), 8-.. F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (9), -5.. F. G. Garvan, Generalizations of Dyson's rank, Ph. D. thesis, Pennsylvania State Univ., 98.

5 DYSON'S CRANK OF A PARTITION 7 5., New combinatorial interpretations of Ramanujan's partition congruences mod 5,7 and, Trans. Amer. Math. Soc. (to appear). DEPARTMENT OF MATHEMATICS, PENNSYLVANIA STATE UNIVERSITY, SITY PARK, PENNSYLVANIA 8 UNIVER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WISCONSIN MADISON, MAD ISON, WISCONSIN 57 Current address (F. G. Garvan): Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 5555

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