G. E. Andrews, private communication.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....partitions of n with crank k. We also show that self pseudo conjugate partitions of n (introduced there) are equinumerous with partitions of n into distinct odd parts. Finally, in Appendix B we outline an alternative proof of the formula (5. 17) This proof was communicated to us by George Andrews [7]. 2. Polynomial analogues of Euler s pentagonal number theorem We say that a partition is in the box [L; M ] if its largest part does not exceed L and the number of parts does not exceed M . In other words, L; M: It is well known [5] that the generating functions for partitions ....

.... 2i (aq ) j (aq j a where a = q , r 0. Since the limit of the sequence fq g is equal to zero, we may treat a in (5.17) as a free parameter. In Appendix B we discuss an alternative proof of (5. 17) This proof was communicated to us by George Andrews [7]. In the past, fundamental as they are, modular representations have not received the attention they deserve. Recently, Alladi [1] used 2 modular representations to provide an elegant combinatorial bijection for a variant of G ollnitz s partition theorem. However, in [1] partitions into only ....

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G. E. Andrews, private communication.

....applications of Rogers [22, p. 29, second eq. nonterminating 6 5 summation (cf. 11, Eq. 2.7.1) and elementary manipulations of series. The method of proof we apply extends that already used by M. Jackson [19, Sec. 4] in her rst elementary proof (as pointed out to us by George Andrews [2]) of Ramanujan s 1 1 summation formula [15] cf. 11, Eq. 5.2.1) Jackson s proof essentially derives the 1 1 summation from the q Gau summation, by manipulation of series. In view of this background, it is surprising that this method has not been further applied for half a century. A ....

G. E. Andrews, private communication, June 2000.

....where the series either terminates, or jzj 1, for convergence. The summation in (7.3) was rst discovered by Cauchy [10] Clearly, 7.2) reduces to (7.3) when b = q. The rst elementary proof of the 1 1 summation formula (7.2) was given by M. Jackson (as pointed out to us by George Andrews [3]) Jackson s proof essentially derives the 1 1 summation from the q Gau summation, by manipulation of series. In [32] we reviewed Jackson s proof and extended it to a method to provide new elementary proofs of Dougall s [14] 2 H 2 summation and Bailey s [7] very well poised 6 6 summation, ....

G. E. Andrews, private communication, June 2000.

....BUFFER. Actually all three variables are located in the same place. We can achieve the same e#ect in C code with a union type as follows: typedef struct byte counter[ 3 ] byte threshold[ 3 ] byte status[ 4 ] generated type 1; typedef struct byte ykm data[ 6 ] word gen info[ 2 ]; generated type 2; typedef union byte local buffer[ 100 ] generated type 1 l3para data1[ 10 ] generated type 2 l3para data2[ 10 ] generated type 3; generated type 3 generated var 1; generated var 1.l3para data1[ 2 ] counter[ 3] 5; This translation is slightly di#erent from that ....

....typedef struct byte ykm data[ 6 ] word gen info[ 2 ] generated type 2; typedef union byte local buffer[ 100 ] generated type 1 l3para data1[ 10 ] generated type 2 l3para data2[ 10 ] generated type 3; generated type 3 generated var 1; generated var 1. l3para data1[ 2 ].counter[ 3] 5; This translation is slightly di#erent from that proposed by Mikkonen [74] It has the advantage that it is more simple to generate from PL M code than Mikkonen s 86 proposal. Note that we have used additional (generated) types and variables. In order to provide correct ....

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Andrews, K., Private communication, October 1997.

....and notes that to obtain (1.4) the variable x is to be replaced by xq and then q 2 by q. With similar substitutions (1.5) can be derived from (1.4) As those substitutions are made within finite expressions the derivations are straightforward. On the other hand, as shown to us by Andrews [An98], and as it is well known in the case m = n, identity (1.4) can be proved by means of the q binomial identity in its finite form. Proceed as follows: x Gamma1 ; q) n (xq; q) m = Gamma1) n x Gamman q n(n Gamma1) 2 (xq 1 Gamman ; q) n (xq; q) m = Gamma1) n x Gamman q ....

G. E. Andrews, Private communication, 1998.

....belongs to. According to the fragments, they can conclude whether the expanded macro should be translated into the body of the macro or into the call of the macro. The same practice can be applied to translating all textual substitutions including conditional compilation. According to the authors [3], the translation of a program is performed in several sessions, and each conditionally compiled program region is translated in different sessions. Thus, some parts of the program are translated several times. 4 Basic idea In this section, we first introduce some assumptions which the basic idea ....

Andrews, K., Private communication (via e-mail), October, 1997.

....belongs to. According to the fragments, they can conclude whether the expanded macro should be translated into the body of the macro or into the call of the macro. The same practice can be applied to translating all textual substitutions including conditional compilation. According to the authors [3], the translation of a program is performed in several sessions, and each conditionally compiled program region is translated in different sessions. Thus, some parts of the program are translated several times. 3 Solution In this section, we first introduce some assumptions which the solution is ....

K. Andrews, Private communication (via e-mail), in October, 1997.

....of any defect d where p d divides n . This is equivalent to the conjecture that for all n the group algebra KS n has a p block of defect zero, see [5] 6] It appears to be folklore that the existence of blocks of defect zero should be true for large n, and this is supported by work of [4] [1], 2] by means of modular forms. We consider an elementary approach which is motivated by representation theory. It is well known that the irreducible characters of S n are parametrized by partitions of n; and moreover the Nakayama conjecture holds: Irreducible characters and belong ....

Andrews, G. E.: private communication

....case) then introduce two further specializations of both triple and quintuple product identities to complete the calculation. It seems that Farkas Kra s identity is much deeper than its previous two sisters. 2. The finite and infinite versions of the triple product As shown to us by Andrews [An98], and as it is well known in the case m = n, identity (1.4) can be proved by means of the q binomial identity in its finite form. Proceed as follows: x Gamma1 ; q) n (xq; q) m = Gamma1) n x Gamman q n(n Gamma1) 2 (xq 1 Gamman ; q) n (xq; q) m = Gamma1) n x Gamman q ....

G. E. Andrews, Private communication, 1998.

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