\input amstex \input amsppt.sty \magnification1200 \hsize13cm \vsize19cm \NoBlackBoxes \def\la{\lambda} \def\suml{\sum\limits} \def\prodl{\prod\limits} \def\liml{\lim\limits} \def\({\left(} \def\){\right)} \head {\bf Three Aspects of Partitions}\endhead \vskip0,5cm \centerline{by} \vskip0,5cm \centerline{\smc George E. Andrews\footnote {Partially supported by National Science Foundation Grant DMS 8702695-03 and the IBM T.~J. Watson Research Center.}} \vskip1,5cm \topmatter \address IBM\newline The Thomas J. Watson Research Center \newline Yorktown Heights, NY 10598 USA \newline and \newline The Pennsylvania State University \newline University Park, PA, PA 16802 USA \newline \endaddress \endtopmatter \document {\bf 1.~Introduction} \vskip0,5cm In this paper we shall discuss three topics in partitions. Section 2 is devoted to partitions with difference conditions and is an elucidation of joint work with J.~B. Olsson \cite {16}. In Section 3 we discuss certain partition problems which have their origins in statistical mechanics. We take as the theme for this section Euler's article: Exemplum Memorabile Inductionis Fallacis \cite {23}. The material for this section is closely related to the work in \cite {10}. Section 4 contains a discussion of some of Ramanujan's formulas from both his Notebooks and lost Notebook. More extensive accounts of this topic are found in \cite {8} and \cite {11}. \vskip1cm {\bf 2.~Partitions with Difference Conditions} \vskip0,5cm The work in this section is based on \cite {16}, joint work with J.~B. Olsson. In 1989, Olsson was studying Mullineux's conjecture \cite {29} which briefly may be described as a "conjugation" map for $p$--regular partitions (i.e\. partitions with no part repeated more than $p-1$ times). As Mullineux asserts \cite {29; p.~60}: "\dots when $p$ is prime it is conjectured that this bijection (i.e\. conjugation) arises in the representation theory of the symmetric group $S_n$ of degree $n$. Farahat, M\"uller and Peel \cite {24} have shown how to form a 'good' labelling of the irreducible $p$--modular representations of $S_n$ (for prime $p$) by $p$--regular partitions of $n$. Now the alternating representation of $S_n$ induces in the usual way a bijection (whose square is the identity) upon these representations and hence induces a similar bijection upon the set of $p$--regular partititons via the labelling. For low values of $n$ this group theoretic bijection agrees with the one constructed here; the verification of this has been carried out using the tables of decomposition numbers found in Kerber and Peel (\cite {27}, $p=3,\ n\le10,\ n\ne 7$); Robinson (\cite {30}, $p=3,\ n=7$) and Wagner (\cite {34}, $p=5,7$, $\le 8$)." Olsson calculated the number of partitions fixed by Mullineux's map and those fixed by the Farahat--M\"uller--Peel \cite {24} induced map. If the two maps are the same then obviously the cardinalities of the two sets of fixed points will be identical. This calculation led to the following: \proclaim{Problem} Let $\lambda =(a_1,a_2,\dots,a_k)$ be a partition of $n$. Consider the two following sets of conditions ($p$ an odd prime). \vskip0,3cm $\cases \text {\rm(1A)}\hskip0,5cm p \not\kern2pt\mid a_i \ \text {and}\ 2\not\kern2pt\mid a_i\ \text {for all}\ i,\\ \text {\rm(1B)}\hskip0,5cm a_i\ne a_{i+1}\ \text {for all}\ i. \endcases$ \vskip0,5cm $\cases \text {\rm(2A)}\hskip0,5cm 2\mid a_i \ \text {if and only if}\ p\mid a_i \ \text {for all}\ i,\\ \text {\rm(2B)}\hskip0,5cm 0\le a_i-a_{i+1}\le 2p\ \text {for all $i$}\ (a_{k+1}=0) \\ \text {\rm(2C)}\hskip0,5cm \text {if}\ a_i=a_{i+1}\ \text {then}\ a_i \ \text {is even}, \\ \text {\rm(2D)}\hskip0,5cm \text {if}\ a_i-a_{i+1}=2p \ \text {then}\ a_i \ \text {is odd}.\endcases$ \endproclaim \vskip0,5cm Is the number of partitions of $n$ satisfying (1A)--(1B) equal to the number of partitions satisfying (2A)--(2D)? The simplest possible case is $p=3$. In this case conditions (1A)--(1B) describe partitions into distinct parts $\equiv 1$ or 5 (mod 6), and this suggests the following result of Schur \cite {32} specialized to fit this instance. \proclaim{Theorem 2.1 \rm\cite {32}} The number or partititons of $n$ into distinct parts congruent to 1 or 5 mod 6 equals the number of partitions of $n$ into parts congruent to 0, 1 or 5 mod 6 with the condition that the difference between parts is at least 6 and greater than 6 between two multiples of 6. \endproclaim For example there are 11 partitions of 36 into distinct parts congruent to 1 or 5 (mod 6): 35+1, 31+5, 29+7, 25+11, 23+13, 23+7+5+1, 19+17, 19+11+5+1, 17+13+5+1, 17+11+7+1, 13+11+7+5. The second class of eleven partitions arising from Schur's Theorem when $n=36$ is: 36, 35+1, 31+5, 30+6, 29+7, 25+11, 24+12, 24+11+1, 23+13, 23+12+1, 19+12+5. In contrast, the eleven partitions arising from conditions (2A)--(2D) in the Problem for $n=36$, $p=3$ are: 17+11+7+1, 13+11+7+5, 13+11+6+5+1, 12+12+7+5, 12+11+7+5+1, 12+11+6+6+1, 12+11+6+6+5, 11+7+6+6+5+1, 11+6+6+6+6+1, 7+6+6+6+6+5, 6+6+6+6+6+5+1. Inspection shows that MacMahon's theory of modular partitions for modulus 6 \cite {28} provides a perfect bijection between these two latter classes of partitions. In MacMahon's representation each part is represented by a row of 6's with the residue mod 6 tacked on at the end. Consequently the eleven MacMahon graphs of the final set of partitions above is: $$\matrix \format \l \hskip1cm & \l \hskip1cm & \l \hskip1cm & \l \hskip1cm & \l \\ 6\quad 6\quad 5 & 6\quad 6\quad 1 & 6\quad 6 & 6\quad 6\quad 1 & 6\quad 6 \\ 6\quad 5 & 6 \quad 5 & 6\quad 6 & 6\quad 5 & 6\quad 5 \\ 6\quad 1 & 6\quad 1 & 6\quad 1& 6\quad 6 & 1 \\ 1 & 5 & 5 & 5 & 5 \\ & & & 1 & 1 \endmatrix$$ $$\matrix \format \l \hskip1cm & \l \hskip1cm & \l \hskip1cm & \l \hskip1cm & \l \hskip1cm & \l\\ 6 \quad 6 & 6\quad 6 & 6\quad 5 & 6\quad 5 & 6\quad 1 & 6 \\ 6\quad 5 & 6\quad 1 & 6\quad 1 & 6 & 6 & 6 \\ 6 & 6 & 6 & 6 & 6 & 6 \\ 6 & 6 & 6 & 6 & 6 & 6 \\ 1 & 5 & 5 & 6 & 6 & 6\\ & & 1 & 1 & 5 & 5 \\ & & & & & 1 \endmatrix$$ Now we form a new set of partitions by reading these graphs by columns instead of rows. The result is Schur's second set of partitions, and a little reflection shows that the above mapping always provides a bijection between Schur's partitions and those of the second kind in the Problem where $p=3$. The relationship described above suggests that the Problem may be solved by relating it to some generalization of Schur's Theorem and then applying MacMahon's modular theory. While there arose some difficulties, this program eventually produced the following result. \proclaim{Theorem 2.2 \rm\cite {16}} Let $A=\{a_1,a_2,\dots,a_r\}$ be a set of $r$ distinct positive integers arranged in increasing order, and let $N$ be an integer larger than $a_r$. Let $P_1(A;N,n)$ denote the number of partitions of $n$ into distinct parts each of which is congruent to some $a_i$ modulo $N$. Let $P_2(A;N,n)$ denote the number of partititons of $n$ into parts each of which is congruent to 0 or to some $a_i$ modulo $N$, in addition only parts divisible by $N$ may be reapeated, the smallest part is $0$. Surprisingly, however, this marvelous rule $$3\binom {m+1} {0}_2-\binom {m+2} {0}_2=F_m (F_m+1),\quad -1\le m\le 7, \tag 3.2$$ is {\it false} for $m>7$. In order to understand (3.2) we define $$E_m(a,b)=\suml _{\lambda=-\infty} ^{\infty} \( \binom {m} {10\lambda+a}_2-\binom {m} {10\lambda+b}_2 \).\tag 3.3$$ As part of Theorem 3.1, we show that $$2E_{m+1}(0,1)=F_m(F_m+1),\tag 3.4$$ from which (3.2) follows by inspection. \proclaim{Theorem 3.1} $$\gather 2E_m(1,2)=2E_{m-1}(0,3)=2E_{m+1}(0,1)=F_m(F_m+1),\tag 3.5\\ E_m(1,4)=E_{m+1}(2,3)=F_{m+1}F_m,\tag 3.6\\ 2E_m(3,4)=2E_{m-1}(2,5)=2E_{m+1}(4,5)=F_m(F_m-1), \tag 3.7\\ 2E_m(1,3)=F_{2m}+F_{m},\tag 3.8\\ 2E_m(2,4)=F_{2m}-F_{m}, \tag 3.9\\ 2E_m(1,5)=F_{2m+1}-F_{m-1}, \tag 3.10\\ 2E_m(0,4)=F_{2m+1}+F_{m-1}, \tag 3.11\\ 2E_m(0,2)=F_{2m-1}+F_{m+1}, \tag 3.12\\ 2E_m(3,5)=F_{2m-1}-F_{m+1}, \tag 3.13\\ E_m(0,5)=F_{2m-1}+F_mF_{m-1}. \tag 3.14 \endgather$$ \endproclaim \demo {Sketch of Proof} We note that $$\gather E_m(a,b)=-E_m(b,a), \quad E_m(10r+a, 10s+b)=E_m(a,b), \tag 3.15\\ E_m(10-a,b)=E_m(a,b), \tag 3.16\\ E_m(10-a,b)=E_m(a,b)=E_m(a,10-b), \tag 3.16 \endgather$$ and that $$E_m(a,b)=E_{m-1}(a,b)+E_{m-1}(a-1,b-1). \tag 3.17$$ Equations (3.15)--(3.17) totally define $E_m(a,b)$ together with appropriate initial values. The rest follows by induction.\quad \quad \qed \enddemo As a corollary of Theorem 3.1 it is easy to show that $$3\binom {m+1} {0}_2-\binom {m+2} {0}_2=2\binom {m+1} {0}_2-2\binom {m+1} {1}_2$$ $$\hskip4,3cm =2E_{m+1}(0,1)\quad \quad (\text {for}\ m\le 7)$$ $$\hskip4,4cm =F_m(F_{m+1}+1)\quad \quad (\text {by (3.5)}).$$ The natural question that arises is: Are there $q$--analogs of at least portions of Theorem 2.1 and if so, what are the implications for the Rogers--Ramanujan type identities? In \cite {10}, $q$--analogs of (3.8)--(3.11) were found. For example, we recall Schur's polynomials $G_1(q)=G_2(q)=1,\ G_n(q)=G_{n-1}(q)+q ^{n-2}G_{n-2}(q)$ for $n>2$. Schur \cite {31} showed that $$ G_{n+1}(q)=\suml _{\lambda=-\infty} ^{\infty}(-1) ^\lambda q^{\lambda (5\lambda +1)/2}\bmatrix n\\ \lfloor \frac {n-5\lambda} {2}\rfloor \endbmatrix_q. \tag 3.18$$ where $\lfloor x \rfloor$ is the greatest integer $\le x$ and $$\bmatrix A\\ B \endbmatrix = \bmatrix A\\ B \endbmatrix_q=\cases \frac {(1-q ^A)(1-q ^{A-1})\cdots (1-q ^{A-B+1})} {(1-q ^B)(1-q ^{B-1})\cdots (1-q)}& 0\le B\le A ,\\ 0&\text {otherwise.}\endcases $$ The $q$--analog of trinomial coefficients appropriate for our discussion here is $$\binom {m;B;q} {A}_2=\suml _{j\ge 0} ^{}q ^{j(j+B)}\bmatrix m\\ j\endbmatrix \bmatrix m-j\\ j+A\endbmatrix. \tag 3.19$$ Note that $$\binom {m;B;1} {A}_2=\binom {m} {A}_2. \tag 3.20$$ Schur [31] deduced the first Rogers-Ramanujan identity as a limiting case of (3.18). For $q$--analogs of (3.11) and (3.10) respectively, we discover that $$\multline \frac {1} {2}(G_{2m+1}(q ^{1/2}) +G_{2m+1}(-q ^{1/2}))\\ =\suml _{\la=-\infty} ^{\infty} q ^{30 \la ^2 - 2 \la} \binom {m;10\la;q} {10\la}_2 - \suml _{\la=-\infty} ^{\infty} q ^{30\la ^2+22\la+4} \binom {m;10 \la +4;q} {10 \la +4}_2, \endmultline\tag 3.21$$ $$\multline \frac {q ^{-1/2}} {2} (G_{2m+1}(q ^{1/2})-G_{2m+1}(-q ^{1/2}))\\ =\suml _{\la=-\infty} ^{\infty} q ^{30\la ^2+8\la} \binom {m;10\la+1;q} {10\la+1}_2 - \suml _{\la=-\infty} ^{\infty} q ^{30\la ^2+32\la+8} \binom {m;10\la+5;q} {10\la+5}_2. \endmultline\tag 3.22$$ In contrast with Schur's identity (3.18), we find [9] that $$G_{n+1}(q)=\suml _{\la=-\infty} ^{\infty} (-1) ^\la q ^{\la(5\la+1)/2} \binom {n;\lfloor \frac {5\la+1} {2} \rfloor;q} {\lfloor \frac {5\la+1} {2} \rfloor}_2. \tag 3.23$$ From (3.23) one can again deduce the first Rogers-Ramanujan identity as a limiting case; however, the simple replacement of the Gaussian polynomial in (3.18) by a $q$-trinomial in (3.23) is at the very least quite surprising. Furthermore the limiting cases of (3.21) and (3.22) do not lead to the first Rogers-Ramanujan identity, but rather to the Rogers-Ramanujan series split into even and odd parts. Namely $$\multline 1+\suml _{j=1} ^{\infty} \frac {q ^{j ^{2}}} {(1-q) (1-q^2) \cdots (1-q^j)}= \frac {1} {\prodl _{n=1} ^{\infty} (1-q ^{2n})}\\ \Biggl \{\suml _{\la =-\infty} ^{\infty} (q ^{60 \la ^2-4 \la}-q ^{60 \la^2 + 44 \la + 8})+ q \suml _{\la=-\infty} ^{\infty}(q ^{60 \la ^2 + 16 \la} -q ^{60 \la ^2 + 64 \la + 16}) \Biggr \}. \endmultline$$ The main riddle concerning all this is precisely the combinatorics. In [3], [4; Ch.~ 9], [15], [21], we see clearly the partition-theoretic significance of (3.18). However the $q$-trinomial version (3.23) is still a complete conbinatorial mystery. \vskip0,6cm {\bf 4.~ Ramanujan} \vskip0,3cm The work in the last several years stemming from Ramanujan's discoveries has truly been amazing. Bruce Berndt at the University of Illinois has been the chief architect of much of the work. He is bringing out edited versions [17], [18], [19] of Ramanujan's famous Notebooks. In addition, the book Ramanujan Revisited [7], edited by Berndt and others, describes recent research on a number of topics related to Ramanujan's work, and the Lost Notebook has been published in photostatic reproduction by Springer--Narosa in 1987. The Mock Theta Conjectures arising from the Lost Notebook were described as follows by Ian Stewart in Nature [33]: \vskip0,3cm {\leftskip1cm\rightskip1cm \noindent One of the most unusual people in the annals of mathematical research is Srinivasa Ramanujan, a self-taught Indian mathematician whose premature death left a rich legacy of unproved theorems. Ramanujan was preeminent in an unfashionable field --- the manipulation of formulas. He tended to state his results without proofs --- indeed on many occasions it is unclear whether he possessed proofs in the accepted sense --- yet he had an uncanny knack of penetrating to the heart of the matter. Over the years, many of Ramanujan's claims have been established in full rigour, although seldom easily. The most recent example, the `mock theta conjectures', is especially striking, because the results in question were stated in Ramanujan's final correspondence with his collaborator Godfrey H. Hardy. The conjectures have recently been proved by Dean Hickerson [26] \newline \dots The proof involves delicate manipulations of infinite series of a kind that would have delighted Ramanujan. The astonishing complexity of the proof underlines, yet again, the depth of Ramanujan's genius. It is very hard to see how anyone could have been led to such results without getting bogged down in the fine detail. Ramanujan was the formula man {\it par exellence}, operating in a period when formulas were out of fashion. Today's renewed emphasis on combinatorics, inspired in a part by the digital nature of computers, has provoked a renewed interest in formula manipulations. The half--forgotten ideas of Srinivasa Ramanujan are breathing new life into number theory and combinatorics. \par} \vskip0,3cm In what follows we provide a sketch of recent work arising from Ramanujan's Notebooks. In [22], D. Bressoud gave a very simple proof of the Rogers--Ramanujan identities. We may for purposes of example slightly rephrase his proof. Namely, he noted that $$\alpha _n= \cases 1 & \text {if} \quad n=0 \\ (-1) ^n \left ( zq^{ \binom {n} {2} } + z ^{-n}q ^{\binom {n+1} {2}} \right) &\text {if}\quad n>0, \endcases \tag 4.1$$ and $$\beta_n= \frac {(z)_n\,(q/z)_n} {(q)_{2n}} \tag 4.2$$ form a Bailey pair [6; p.~26], i.e\. $$\beta_n=\suml _{j=0} ^{n} \frac {\alpha_j} {(q)_{n-j} \,(aq)_{n+j}} \tag 4.3$$ where $$(A)_n=(A;q)_n=(1-A)(1-Aq) \cdots (1-Aq ^{n-1}). \tag 4.4$$ A weak iterated version of Bailey's Lemma asserts that if $\alpha_n$ and $\beta_n$ form a Bailey pair, then $$\frac {1} {(aq;q)_\infty} \suml _{n=0} ^{\infty} q ^{kn ^2} a ^{kn} \alpha _n = \suml _{n_k \geq \cdots \geq n_1 \geq 0 } ^{} \frac {\alpha ^{n_1+ \cdots + n_ a} q ^{n ^2_1+ \cdots + n ^2_k} \beta_{n_1}} {(q)_{n_k-n_{k-1}} (q)_{n_{k-1}-n_{k-2}} \cdots (q)_{n_2-n_1}}. \tag4.5$$ Bressoud's proof [22] can be viewed as setting $z=1$ in (4.1) and (4.2) and inserting the resulting pair in (4.5) with $k=2$. \newline If instead, we take (4.1) and (4.2) as they are and insert them into (4.5) with $k=1$ and $\alpha=1$, we find $$\align \suml _{n=0} ^{\infty} \frac {(z)_n\,(q/z)_n \,q ^{n ^2}} {(q)_{2n}}&= \frac {1 + \suml _{n=1} ^{\infty} (-1) ^n \left(z ^n q ^{\binom {n} {2}} + z ^{-n} q ^{ \binom {n+1} {2}} \right)q ^{n ^2}} {(q)_\infty} \\ &=\frac {\suml _{n=-\infty} ^{\infty} (-z) ^n q ^{n(3n-1)/2}} {(q)_\infty} \tag 4.6 \\ &=\frac {(q ^3;q ^3)_\infty (zq;q ^3)_\infty (z ^{-1}q ^2;q ^3)_\infty} {(q)_\infty }, \endalign$$ a result from the Lost Notebook. If we differentiate each entry in (4.6) and then set $z=1$, we deduce $$ \align \suml _{n=1} ^{\infty} \frac {(1-q) (1-q ^2) \cdots (1-q ^{n-1}) q ^{n ^2}} {(1-q ^{n+1}) (1-q ^{n+2}) \cdots (1-q ^{2n})} &= \frac {\suml _{n=-\infty} ^{\infty} (-1) ^n nq ^{n(3n-1)/2}} {(q)_\infty} \\ &=\suml _{n=0} ^{\infty} \Biggl (\frac {q ^{3n+1}} {1-q ^{3n+1}} - \frac {q ^{3n+2}} {1-q ^{3n+2}} \Biggr ) \\ &=\suml _{n=1} ^{\infty} q ^n \Biggl (\suml _{d/n} ^{} \biggl (\frac {d} {3} \biggr ) \Biggr ) , \tag 4.7 \endalign $$ where $\left(\frac {d} {3} \right)$ is the Legendre symbol. Thus just beneath the surface of (4.6) is a $q$-series (namely the left--hand side of (4.7)) with multiplicative coefficients all non--negative, all indeed $O(n^2)$. This suggests that the underlying combinatorics of (4.7) is well worth a look, and we shall not be disappointed. Following the program outlined in [5], we consider $$f(t,q)=\suml _{n=1} ^{\infty} \frac {t ^{2n}q ^{n ^2}(-tq)_{n-1}} {(t)_{n+1}\,(t ^2q;q ^2)_n} . \tag 4.8$$ The function $f(t,q)$ was constructed to both satisfy a first order nonhomogeneous $q$-difference equation $$f(t,q)=\frac {t ^2q} {(1-t) (1-tq) (1-t ^2q)} + \frac {t ^2q(1+tq)} {(1-t) (1-t ^2q)} f(tq,q), \tag 4.9$$ and to reduce in the case $t=-1$ to the left--hand side of (4.7): $$f(-1,q)=\frac {1} {2} \suml _{n=1} ^{\infty} \frac {q ^{n ^2} (q)_{n-1}} {(q ^{n+1};q)_n}.$$ If the magic of Ramanujan's mathematics is operating here, then $f(t,q)$ should be an interesting generating function of polynomials (in $q$), and $f(q,q ^2)$, $\liml _{t\to 1^-} f(t,q) (1-t)$ should also exhibit interesting structure. In this regard, we find $$f(t,q)=\suml _{n=2} ^{\infty} t ^np_n(q)$$ with $$p_n(q)=\suml _{\la=-\infty} ^{\infty} q ^{(2\la+1)(3\la+1)} \bmatrix n \\ \lfloor \frac {n} {2} \rfloor -3 \la -1 \endbmatrix,$$ $$\liml _{t \to 1^-} (1-t) f (t,q)= \suml _{n=1} ^{\infty} \frac {(1+q) ^2 (1+q ^2) ^2 \cdots (1+q ^{n-1}) ^2 (1+q ^n) q ^{n ^2}} {(1-q) (1-q ^2) \cdots (1-q ^{2n})} $$ $$=q \prodl _{n=1} ^{\infty} \frac {(1-q ^{12n}) (1+q ^{12n-1}) (1+q ^{12n-11})} {(1-q ^n)},$$ and finally $$\frac {1} {1-q} + (1+q)f(q,q ^2) =\suml _{n=0} ^{\infty} \frac {q ^{2n ^2+2n} (-q;q ^2)_n} {(q)_{2n+1} (-q ^2;q ^2)_n} =\prodl _{n=1} ^{\infty} \frac {(1-q ^{6n}) (1+q ^{6n-1}) (1+q ^{6n-5})} {(1-q ^{2n})}.$$ It should be added that the above discoveries all resulted from a consideration of four seemingly benign identities of Ramanujan [11]. The simplest of which is $$\suml _{n=1} ^{\infty} \frac {(-1) ^{n-1}nq ^{\binom {n+2} {2}} (1-q ^n)} {(1+q ^n) ^2}= \biggl (\suml _{n=-\infty} ^{\infty} (-1) ^nq ^{n ^2} \biggr ) ^2 \suml _{n=1} ^{\infty} \frac {nq ^{\binom {n+1} {2}}} {1-q ^n}.$$ The proof of this result relies in an essential way on Bressoud's Bailey pair (4.1), (4.2) differentiated with respect to $z$ and with $z$ then set equal to $1$ together with a general $q$-hypergeometric identity of Bailey [25, p.~42, eq.~(2.10.10)]. \vskip0,5cm {\bf 5.~ Conclusion} \vskip0,3cm These lectures being expository lack the details necessary for a full understanding of the underlying proofs. In this regard, Section 2 is an exposition of [16]; Section 3 of [10], and Section 4 of [5], [8] and [11]. Related background material may be found in [2] and [4] for Section 2, [9] for Section 3 and [6; Ch.9] for Section 4. \vskip0,5cm \Refs \ref\no 1 \by ALLADI K. and GORDON B. \paper Partition identities and a continued fraction of Ramanujan \jour J. Combin\. Theory Ser.~A\vol 63\pages 275--300\endref \ref\no 2 \by ANDREWS G.E. \paper A general theorem on partitions with difference conditions \jour Amer\. J. Math.\vol 91 \yr 1969 \pages 18--24 \endref \ref\no 3 \by ANDREWS G.E\. \paper Sieves in the theory of partitions \jour Amer\. J. Math\. \vol 94 \yr 1972 \pages 1214--1230 \endref \ref\no 4 \by ANDREWS G.E. \book The Theory of Partitions \publ Encycl. Math\. and Its Application, Vol.~2, G.-C.~Rota, ed.\publaddr Addison-Wesley, Reading\yr 1976 \finalinfo (Reissued: Cambridge University Press, London and New York, 1985) \endref \ref\no 5 \by ANDREWS G.E. \book Combinatorics and Ramanujan's ``lost" notebook \publ London Math\. Soc\. Lecture Series, Vol.~103 \publaddr Cambridge University Press, London \yr 1985 \pages 1--23 \endref \ref\no 6 \by ANDREWS G.E. \book $q$-Series \publ C.B.M.S Series in Math., Vol.~66\publaddr Amer\. Math\. Soc\., Providence \yr 1986 \endref \ref\no 7 \by ANDREWS G.E. \paper Ramanujan Revisited \jour Academic Press, New York \yr 1988 \endref \ref\no 8 \by ANDREWS G.E. \paper Mock theta functions \jour Proc. Symp\. in Pure Math\. \vol 49 \yr 1989 \pages 283--298 \endref \ref\no 9 \by ANDREWS G.E. \paper $q$-Trinomial coefficients and the Rogers-Ramanujan identities \inbook Analytic Number Theory\eds Berndt et al. \publ Birkh\"auser\publaddr Boston \yr 1990 \pages 1--11 \endref \ref\no 10 \by ANDREWS G.E. \paper Euler's ``Exemplum Memorabile Inductionis Fallacis" and $q$-trinomial coefficients \jour J. Amer\. Math\. Soc\. \vol 3 \yr 1990 \pages 653--669 \endref \ref\no 11 \by ANDREWS G.E. \paper Bailey chains and generalized Lambert series: I. four identities of Ramanujan \jour Illinois J. Math\.\vol 36\pages 251--274\endref \ref\no 12 \by ANDREWS G.E. and BAXTER R.J. \paper Lattice gas generalizations of the hard hexagon model: I. star triangle relations and local densities \jour J. Stat\. Phys\. \vol 44 \yr 1986 \pages 259--271 \endref \ref\no 13 \by ANDREWS G.E. and BAXTER R.J. \paper Lattice gas generalizations of the hard hexagon model: II. the local densities as elliptic functions \jour J. Stat\. Phys\. \vol 44 \yr 1986 \pages 713--728 \endref \ref\no 14 \by ANDREWS G.E. and BAXTER R.J. \paper Lattice gas generalizations of the hard hexagon model: III $q$-trinomial coefficients \jour J. Stat\. Phys\. \vol 47 \yr 1987 \pages 297--230 \endref \ref\no 15 \by ANDREWS G.E., BAXTER R.J., BRESSOUD D.M., BURGE W.H., FORRESTER P.J. and VIENNOT G. \paper Partitions with prescribed hook differences \jour Europ\. J. Math\. \vol 8 \yr 1987 \pages 341--350 \endref \ref\no 16 \by ANDREWS G.E. and OLSSON J.B. \paper Partition identities with an application to group representation theory \jour J. Reine und angew\. Math\. \vol 413 \yr 1991 \pages 198--212 \endref \ref\no 17 \by BERNDT B.C. \book Ramanujan's Notebooks, Part I \publ Springer Verlag\publaddr New York and Berlin \yr 1985 \endref \ref\no 18 \by BERNDT B.C. \book Ramanujan's Notebooks, Part II \publ Springer Verlag\publaddr New York and Berlin \yr 1990 \endref \ref\no 19 \by BERNDT B.C. \book Ramanujan's Notebooks, Part III \publ Springer Verlag\publaddr New York and Berlin \yr 1991\endref \ref\no 20 \by BESSENRODT C. \paper A combinatorial proof of a refinement of the Andrews-Olsson partition identity \jour Europ\. J. Combin\.\vol 12\yr 1991\pages 271--276\endref \ref\no 21 \by BRESSOUD D.M. \paper Extension of the partition sieve \jour J. Number Th\. \vol 12 \yr 1980 \pages 87--100 \endref \ref\no 22 \by BRESSOUD D.M. \paper An easy proof of the Rogers-Ramanujan identities \jour J. Number Th\. \vol 16 \yr 1983 \pages 235--241 \endref \ref\no 23 \by EULER L. \paper Observations analyticae \jour Novi Commentarii Academiae Scientarum Petropolitanae \vol 11 \yr 1765 \pages 124--143 \finalinfo also in {\it Opera Omnia, Series 1} Vol.~15, Teubner, pp.~50--69 \endref \ref\no 24 \by FARAHAT H.K., M\"ULLER W. and PEEL M.H \paper The modular characters of the symmetric groups \jour J. Algebra \vol 40 \yr 1976 \pages 354--363 \endref \ref\no 25 \by GASPER G. and RAHMAN M. \book Basic Hypergeometric Series \publ Encycl\. Math\. and Its Applications, Vol.~35, G.-C.~Rota, ed. \publaddr Cambridge University Press, Cambridge\yr 1990 \endref \ref\no 26 \by HICKERSON D.R. \paper A proof of the mock theta conjectures \jour Invent. math \vol 94 \yr 1989 \pages 639--660 \endref \ref\no 27 \by KERBER A. and PEEL M.H. \paper On the decomposition numbers of symmetric and alternating groups \jour Mitt. Math. Sem. Giessen \vol 91 \yr 1971 \pages 45--81 \endref \ref\no 28 \by MACMAHON P.A. \paper The theory of modular partitions \jour Proc. Cambridge Phil. Soc. \vol 21 \yr 1923 \pages 197--204 \finalinfo (Reprinted in the Coll. Papers of P.A. MacMahon, Vol.~1, MIT Press, Cambridge, 1978) \endref \ref\no 29 \by MULLINEUX G. \paper Bijections of $p$-regular partitions and $p$-modular irreducibles of the symmetric groups \jour J. London Math. Soc. (2) \vol 20 \yr 1979 \pages 60--66 \endref \ref\no 30 \by ROBINSON G. de B. \book Representation Theory of the Symmetric Group \publ Edinburgh University Press \yr 1961 \endref \ref\no 31 \by SCHUR I. \paper Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbr\"uche \jour S.-B. Preuss\. Akad. Wiss. Phys.-Math\. Kl. \yr 1917 \pages 302--321 \finalinfo (Reprinted in I. Schur, Gesammelte Abhandlungen, Vol.~2, Springer, Berlin, 1973, pp.~117--136)\endref \ref\no 32 \by SCHUR I. \paper Zur additiven Zahlentheorie \jour S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. \yr 1926 \pages 499--495 \finalinfo (Reprinted in I. Schur, Gesammelte Abhandlungen, Vol.~3 Springer, Berlin, 1973, pp.~43--50) \endref \ref\no 33 \by STEWART I. \paper Mock theta conjectures \jour Nature \vol 339 \yr 1989 \pages 341 \endref \ref\no 34 \by WAGNER A. \paper The faithful representations of the least degree of $S_n$ and $A_n$ over a field of odd characteristic \jour Math. Z. \vol 154 \yr 1976 \pages 103--114 \endref \vskip0,3cm \endRefs \enddocument