449 research outputs found
Anti-lecture Hall Compositions and Overpartitions
We show that the number of anti-lecture hall compositions of n with the first
entry not exceeding k-2 equals the number of overpartitions of n with
non-overlined parts not congruent to modulo k. This identity can be
considered as a refined version of the anti-lecture hall theorem of Corteel and
Savage. To prove this result, we find two Rogers-Ramanujan type identities for
overpartition which are analogous to the Rogers-Ramanjan type identities due to
Andrews. When k is odd, we give an alternative proof by using a generalized
Rogers-Ramanujan identity due to Andrews, a bijection of Corteel and Savage and
a refined version of a bijection also due to Corteel and Savage.Comment: 16 page
The Method of Combinatorial Telescoping
We present a method for proving q-series identities by combinatorial
telescoping, in the sense that one can transform a bijection or a
classification of combinatorial objects into a telescoping relation. We shall
illustrate this method by giving a combinatorial proof of Watson's identity
which implies the Rogers-Ramanujan identities.Comment: 11 pages, 5 figures; to appear in J. Combin. Theory Ser.
A combinatorial proof of the Rogers-Ramanujan and Schur identities
We give a combinatorial proof of the first Rogers-Ramanujan identity by using
two symmetries of a new generalization of Dyson's rank. These symmetries are
established by direct bijections.Comment: 12 pages, 5 figures; incorporated referee suggestions, simplified
definition of (k,m)-rank, to appear in JCT(A
The Rogers-Ramanujan-Gordon Theorem for Overpartitions
Let be the number of partitions of with certain difference
condition and let be the number of partitions of with certain
congruence condition. The Rogers-Ramanujan-Gordon theorem states that
. Lovejoy obtained an overpartition analogue of the
Rogers-Ramanujan-Gordon theorem for the cases and . We find an
overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general
case. Let be the number of overpartitions of satisfying
certain difference condition and be the number of overpartitions
of whose non-overlined parts satisfy certain congruences condition. We show
that . By using a function introduced by Andrews, we
obtain a recurrence relation which implies that the generating function of
equals the generating function of . We also find a
generating function formula of by using Gordon marking
representations of overpartitions, which can be considered as an overpartition
analogue of an identity of Andrews for ordinary partitions.Comment: 26 page
Overpartitions, lattice paths and Rogers-Ramanujan identities
We extend partition-theoretic work of Andrews, Bressoud, and Burge to
overpartitions, defining the notions of successive ranks, generalized Durfee
squares, and generalized lattice paths, and then relating these to
overpartitions defined by multiplicity conditions on the parts. This leads to
many new partition and overpartition identities, and provides a unification of
a number of well-known identities of the Rogers-Ramanujan type. Among these are
Gordon's generalization of the Rogers-Ramanujan identities, Andrews'
generalization of the G\"ollnitz-Gordon identities, and Lovejoy's ``Gordon's
theorems for overpartitions.
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