403 research outputs found
Enumeration of Stack-Sorting Preimages via a Decomposition Lemma
No full textWe give three applications of a recently-proven "Decomposition Lemma," which
allows one to count preimages of certain sets of permutations under West's
stack-sorting map . We first enumerate the permutation class
, finding a new example
of an unbalanced Wilf equivalence. This result is equivalent to the enumeration
of permutations sortable by , where is the bubble
sort map. We then prove that the sets ,
,
and are
counted by the so-called "Boolean-Catalan numbers," settling a conjecture of
the current author and another conjecture of Hossain. This completes the
enumerations of all sets of the form
for
with the exception of the set
. We also find an explicit formula for
, where
is the set of permutations in with descents.
This allows us to prove a conjectured identity involving Catalan numbers and
order ideals in Young's lattice
Binary Codes and Period-2 Orbits of Sequential Dynamical Systems
No full textLet be the (global) SDS map of a sequential dynamical system
(SDS) defined over the complete graph using the update order
in which all vertex functions are equal to the same function . Let denote the maximum number of periodic
orbits of period that an SDS map of the form can have. We
show that is equal to the maximum number of codewords in a binary code
of length with minimum distance at least . This result is significant
because it represents the first interpretation of this fascinating
coding-theoretic sequence other than its original definition
Convergence of monomial expansions in banach spaces
No full text[EN] If E is a Banach sequence space, then each holomorphic function defines a formal power series ¿ ¿ c ¿(f) z ¿. The problem of when such an expansion converges absolutely and actually represents the function goes back to the very beginning of the theory of holomorphic functions on infinite-dimensional spaces. Several very deep results have been given for scalar-valued functions by Ryan, Lempert and Defant, Maestre and Prengel. We go on with this study, looking at monomial expansions of vector-valued holomorphic functions on Banach spaces. Some situations are very different from the scalar-valued case. © 2011 Published by Oxford University Press. All rights reserved.Both authors were supported by the MEC Project MTM2008-03211. The second cited author was partially supported by grants PR2007-0384 (MEC) and UPV-PAID-00-07.Defant, A.; Sevilla Peris, P. (2012). Convergence of monomial expansions in banach spaces. Quarterly Journal of Mathematics. 63(3):569-584. https://doi.org/10.1093/qmath/haq053S56958463
Proofs of Conjectures about Pattern-Avoiding Linear Extensions
No full textAfter fixing a canonical ordering (or labeling) of the elements of a finite
poset, one can associate each linear extension of the poset with a permutation.
Some recent papers consider specific families of posets and ask how many linear
extensions give rise to permutations that avoid certain patterns. We build off
of two of these papers. We first consider pattern avoidance in -ary heaps,
where we obtain a general result that proves a conjecture of Levin, Pudwell,
Riehl, and Sandberg in a special case. We then prove some conjectures that
Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan made about pattern-avoiding
linear extensions of rectangular posets
Postorder Preimages
No full textGiven a set of decreasing plane trees and a permutation , how many
trees in have as their postorder? Using combinatorial and geometric
constructions, we provide a method for answering this question for certain sets
and all permutations . We then provide applications of our results to
the study of the deterministic stack-sorting algorithm
A novel harmonic solution for chatter stability of time periodic systems
No full textChatter vibrations strongly limit productivity in milling. Due to the presence of rotating parts with asymmetric stiffness and stability enhancement strategies which act through a periodic variation of stiffness, there is growing interest in estimating the stability maps of systems with Linear Time Periodic dynamics together with periodic cutting excitation. Applying Exponentially Periodic Modulated test signals to the dynamic cutting force equation and representing the dynamics of the system through the Harmonic Transfer Function, the innovative Harmonic Solution (HS) and its zero-order approximation were derived in this research. HS is a frequency domain representation of a system described by the combination of two independent periodicities. It is possible to take into account these periodicities together in HS or singularly, resulting in the Zero Order HS or in the well-known Multi-Frequency Solution. This novel formulation can deal efficiently with spindle dependent and independent dynamics and is prone to industrial applications due to its flexibility and efficiency. More specifically, in this work the developed methodologies were used to assess the cutting stability of systems with a periodically modulated stiffness. The accuracy and efficiency of HS were validated by comparison with the results achieved by the use of the semi-discretization method. Results are in agreement with those obtained using semi-discretization. Moreover, admitting a slight precision loss, HS and its zero-order approximation are orders of magnitude faster than semi-discretization, giving reliable stability maps from seconds to a few minutes
A Short Proof to Defant and Kravitz's theorem on the Length of Hitomezashi Loops
No full textWe provide a shorter proof to Defant and Kravitz's theorem (arXiv:2201.03461,
Theorem 1.2) on the length of Hitomezashi loops modulo 8.Comment: 4 pages, 2 figure
Boolean, Free, and Classical Cumulants as Tree Enumerations
No full textDefant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called troupes. Using a generalization of troupes that we call weighted troupes, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.18 pages, 7 figure
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