16,040 research outputs found
Lecture Hall Theorems, q-series and Truncated Objects
We show here that the refined theorems for both lecture hall partitions and
anti-lecture hall compositions can be obtained as straightforward consequences
of two q-Chu Vandermonde identities, once an appropriate recurrence is derived.
We use this approach to get new lecture hall-type theorems for truncated
objects. We compute their generating function and give two different
multivariate refinements of these new results : the q-calculus approach gives
(u,v,q)-refinements, while a completely different approach gives odd/even
(x,y)-refinements. From this, we are able to give a combinatorial
characterization of truncated lecture hall partitions and new finitizations of
refinements of Euler's theorem
Incompleteness of relational simulations in the blocking paradigm
Refinement is the notion of development between formal specifications For specifications given in a relational formalism downward and upward simulations are the standard method to verify that a refinement holds their usefulness based upon their soundness and joint completeness This is known to be true for total relational specifications and has been claimed to hold for partial relational specifications in both the non-blocking and blocking interpretations
In this paper we show that downward and upward simulations in the blocking interpretation where domains are guards are not Jointly complete This contradicts earlier claims in the literature We illustrate this with an example (based on one recently constructed by Reeves and Streader) and then construct a proof to show why Joint completeness fails in general (C) 2010 Elsevier B V All rights reserve
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
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