5,707 research outputs found
Proof of a conjectured q,t-Schr\"{o}der identity
A conjecture of Chunwei Song on a limiting case of the q,t-Schr\"{o}der
theorem is proved combinatorially. The proof matches pairs of tableaux to
Catalan words in a manner that preserves differences in the maj statistic.Comment: 8 pages; v2 corrects presentation error in example and notation
(substance of proof unchanged
The part-frequency matrices of a partition
A new combinatorial object is introduced, the part-frequency matrix sequence
of a partition, which is elementary to describe and is naturally motivated by
Glaisher's bijection. We prove results that suggest surprising usefulness for
such a simple tool, including the existence of a related statistic that
realizes every possible Ramanujan-type congruence for the partition function.
To further exhibit its research utility, we give an easy generalization of a
theorem of Andrews, Dixit and Yee on the mock theta functions. Throughout, we
state a number of observations and questions that can motivate an array of
investigations.Comment: Presented at the Kliakhandler Conference 2015, Algebraic
Combinatorics and Applications, at Michigan Technological University. To
appear in the Proceeding
Partitions into a small number of part sizes
We study , the number of partitions of into part sizes, and
find numerous arithmetic progressions where and take on values
divisible by 2 and 4. Expanding earlier work, we show for (A,B) = (36,30), (72,42), (252,114), (196,70), and likely many
other progressions for which our method should easily generalize. Of some
independent interest, we prove that the overpartition function in the first three progressions (the fourth is known), and
thereby show that in each of these progressions
as well, and discuss the relationship between these congruences in more
generality. We end with open questions in this area.Comment: 11 pages; v2, small correction to proof of Theorem 7; v3, clean up
some explanations, acknowledge recent results from Xinhua Xiong on
overpartitions mod 16; v4, final journal version to appear International
Journal of Number Theory (Feb. 2017
A Bijection for Partitions with Initial Repetitions
A theorem of Andrews equates partitions in which no part is repeated more
than 2k-1 times to partitions in which, if j appears at least k times, all
parts less than j also do so. This paper proves the theorem bijectively, with
some of the generalizations that usually arise from such proofs.Comment: 5 page
Congruences for 9-regular partitions modulo 3
It is proved that the number of 9-regular partitions of n is divisible by 3
when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An
infinite family of congruences mod 3 holds in other progressions modulo powers
of 4 and 5. A collection of conjectures includes two congruences modulo higher
powers of 2 and a large family of "congruences with exceptions" for these and
other regular partitions mod 3.Comment: 7 pages. v2: added citations and proof of one conjecture from a
reader. Submitted versio
Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities
In this paper, we give a conjecture, which generalises Euler's partition
theorem involving odd parts and different parts for all moduli. We prove this
conjecture for two family partitions. We give -difference equations for the
related generating function if the moduli is three. We provide new companions
to Rogers-Ramanujan-Andrews-Gordon identities under this conjecture.Comment: 12 pages, revised versio
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