scholarly journals Combinatorics of the Free Baxter Algebra

10.37236/1043 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Marcelo Aguiar ◽  
Walter Moreira
Keyword(s):  
Explicit Description ◽  
Combinatorial Objects ◽  
Linear Basis ◽  
Alternative Description ◽  
Generating Series ◽  
Schröder Paths ◽  
Motzkin Paths

We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of trees and more familiar combinatorial objects including certain Schröder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday's dendriform trialgebras and dialgebras. We show that the free dendriform trialgebra (respectively, dialgebra) on one generator embeds in the free Baxter algebra with a quasi-idempotent map (respectively, with a quasi-idempotent map and an idempotent generator). This refines results of Ebrahimi-Fard and Guo.

scholarly journals ON THE NUMBER OF ABELIAN BORDERED WORDS (WITH AN EXAMPLE OF AUTOMATIC THEOREM-PROVING)

2014 ◽  
Vol 25 (08) ◽  
pp. 1097-1110 ◽  
Author(s):  
DANIEL GOČ ◽  
NARAD RAMPERSAD ◽  
MICHEL RIGO ◽  
PAVEL SALIMOV
Keyword(s):  
Theorem Proving ◽  
Triangular Lattice ◽  
Automatic Theorem Proving ◽  
Combinatorial Objects ◽  
Letter Alphabet ◽  
Automatic Sequences ◽  
Motzkin Paths ◽  
Automatic Theorem

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in ℤ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a 3-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue–Morse word using some kind of automatic theorem-proving provided by a logical characterization of the k-automatic sequences.

scholarly journals Constructing combinatorial operads from monoids

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Rooted Trees ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
International Audience ◽  
Motzkin Paths ◽  
Planar Rooted Trees

International audience We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

scholarly journals Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace

10.37236/7799 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Veronika Irvine ◽  
Stephen Melczer ◽  
Frank Ruskey
Keyword(s):  
Mathematical Model ◽  
Generating Functions ◽  
Finite Lattice ◽  
Lattice Paths ◽  
Change Direction ◽  
Quarter Plane ◽  
Directed Paths ◽  
Schröder Paths ◽  
Motzkin Paths

Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that vertical steps $(\uparrow, \downarrow)$ cannot be consecutive. The set $\mathfrak{A}$ is the union of the well known Motzkin step vectors $\mathfrak{M}=$$\{\rightarrow,$ $\nearrow,$ $\searrow\}$ with the vertical steps $\{\uparrow, \downarrow\}$. An explicit bijection $\phi$ between the exhaustive set of vertically constrained paths formed from $\mathfrak{A}$ and a bisection of the paths generated by $\mathfrak{M}S$ is presented. In a similar manner, paths with the step vectors $\mathfrak{B}=$$\{\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$, the union of Dyck step vectors and constrained vertical steps, are examined.  We show, using the same $\phi$ mapping, that there is a bijection between vertically constrained $\mathfrak{B}$ paths and the subset of Motzkin paths avoiding horizontal steps at even indices.  Generating functions are derived to enumerate these vertically constrained, partially directed paths when restricted to the half and quarter-plane.  Finally, we extend Schröder and Delannoy step sets in a similar manner and find a bijection between these paths and a subset of Schröder paths that are smooth (do not change direction) at a regular horizontal interval.

A Relation Between Schröder Paths and Motzkin Paths

2020 ◽  
Vol 36 (5) ◽  
pp. 1489-1502
Author(s):  
Lin Yang ◽  
Sheng-Liang Yang
Keyword(s):  
Schröder Paths ◽  
Motzkin Paths

scholarly journals Motzkin Paths, Motzkin Polynomials and Recurrence Relations

10.37236/4781 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Roy Oste ◽  
Joris Van der Jeugt
Keyword(s):  
Recurrence Relations ◽  
Catalan Numbers ◽  
Tridiagonal Matrix ◽  
Efficient Computation ◽  
Combinatorial Objects ◽  
Motzkin Numbers ◽  
The Matrix ◽  
Motzkin Paths

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers.  Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce some properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof.

scholarly journals Combinatorial Proofs of Addition Formulas

10.37236/4793 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Xiang-Ke Chang ◽  
Xing-Biao Hu ◽  
Hongchuan Lei ◽  
Yeong-Nan Yeh
Keyword(s):  
Hankel Matrix ◽  
Direct Consequence ◽  
Combinatorial Proof ◽  
Addition Formula ◽  
Polynomial Sequence ◽  
Schröder Paths ◽  
Motzkin Paths ◽  
Addition Formulas

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schröder paths.

The Movement of Ezekiel’s “Living Beings” in 4Q405 Shirot ‘Olat Hashabbat. Part II: The Twelfth Song

2018 ◽  
Vol 27 (1) ◽  
Author(s):  
Annette Evans
Keyword(s):  
Explicit Description ◽  
Plural Form ◽  
The Law ◽  
Mount Sinai

In this article descriptions of angelic movement in the Twelfth Song are compared to descriptions of such activity arising from the throne of God in Ezekiel’s vision in Ezekiel 1 and 10, and to that in the Seventh Song as contained in scroll 4Q403. The penultimate Twelfth Song of the Songs of the Sabbath Sacrifice culminates in a more explicit description of angelic messenger activity and in other nuances. The Twelfth Song was intended to be read on the Sabbath immediately following Shavu’ot, when the traditional synagogue reading is Ezekiel 1 and Exodus 19–20. The possible significance for the author of Songs of the Sabbath Sacrifice of the connection between the giving of the Law at Mount Sinai and Ezekiel’s vision where merkebah thrones and seats appear in the plural form is considered in the conclusion

scholarly journals A symmetric Bloch–Okounkov theorem

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Jan-Willem M. van Ittersum
Keyword(s):  
Symmetric Functions ◽  
Graded Algebra ◽  
Symmetric Polynomials ◽  
Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

scholarly journals SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1074
Author(s):  
Vincent Wagner ◽  
Nicole Erika Radde
Keyword(s):  
Reaction System ◽  
Stochastic Simulation Algorithm ◽  
Test Bed ◽  
Biochemical Reaction Networks ◽  
Full System ◽  
Catalytic Systems ◽  
Reaction Systems ◽  
Sample Paths ◽  
Alternative Description ◽  
Linear Differential

The Chemical Master Equation is a standard approach to model biochemical reaction networks. It consists of a system of linear differential equations, in which each state corresponds to a possible configuration of the reaction system, and the solution describes a time-dependent probability distribution over all configurations. The Stochastic Simulation Algorithm (SSA) is a method to simulate sample paths from this stochastic process. Both approaches are only applicable for small systems, characterized by few reactions and small numbers of molecules. For larger systems, the CME is computationally intractable due to a large number of possible configurations, and the SSA suffers from large reaction propensities. In our study, we focus on catalytic reaction systems, in which substrates are converted by catalytic molecules. We present an alternative description of these systems, called SiCaSMA, in which the full system is subdivided into smaller subsystems with one catalyst molecule each. These single catalyst subsystems can be analyzed individually, and their solutions are concatenated to give the solution of the full system. We show the validity of our approach by applying it to two test-bed reaction systems, a reversible switch of a molecule and methyltransferase-mediated DNA methylation.

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