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### On posets of m-ary words

Gustav Burosch, Hans-Dietrich O.F. Gronau, Jean-Marie Laborde, Ingo Warnke
1996 Discrete Mathematics
f8 m • n , which consists of the naturally ordered subwords ofthe cyclic word on length n on an alphabet of m letters, where subwords are obtained by deleting letters, is introduced and studied.  ...  This poset is of special interest since it is strongly related to several different structured posets, like Boolean lattices, chains, Higman orders and Kruskal-Katona posets.  ...  The partially ordered set of all subwords of the m-ary word Um,n,i = i (i + l)(i + 2) ...  ...

### Counting smaller elements in the Tamari and m-Tamari lattices [article]

Viviane Pons, Grégory Chatel
2015 arXiv   pre-print
We explain how the m-Tamari lattices can be interpreted in terms of m+1-ary trees or a certain class of binary trees.  ...  We then use the interval-posets to recover the functional equation of m-Tamari intervals and to prove a generalized formula that counts the number of elements smaller than or equal to a given tree in the  ...  The number of nodes on the border of a m-binary tree is the same as on its associated (m + 1)-ary tree and still corresponds to the number of touch points of the m-ballot path.  ...

### On a quasi-ordering on Boolean functions [article]

Miguel Couceiro, Maurice Pouzet
2006 arXiv   pre-print
It was proved few years ago that classes of Boolean functions definable by means of functional equations EFHH, or equivalently, by means of relational constraints Pi2, coincide with initial segments of  ...  the quasi-ordered set (Ω, ≤) made of the set Ω of Boolean functions, suitably quasi-ordered.  ...  If f is an n-ary Boolean function and g 1 , . . . , g n are m-ary Boolean functions, then their composition is the m-ary Boolean function f (g 1 , . . . , g n ), whose value on every a ∈ B m is f (g 1  ...

### The Number of Irreducible Polynomials and Lyndon Words with Given Trace

F. Ruskey, C. R. Miers, J. Sawada
2001 SIAM Journal on Discrete Mathematics
This same approach is used to count Lq(n, t), the number of q-ary Lyndon words whose characters sum to t mod q.  ...  This number is given by Lq(n, t) = ( gcd(d, q)µ(d)q n/d )/(qn), where the sum is over all divisors d of n for which gcd(d, q)|t. Both results rely on a new form of Möbius inversion.  ...  Let L q (n, t) denote the number of q-ary Lyndon words of length n and trace t mod q.  ...

### Symmetric chain decomposition of necklace posets [article]

Vivek Dhand
2012 arXiv   pre-print
A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains.  ...  If P is any symmetric chain order, we prove that P^n/Z_n is also a symmetric chain order, where Z_n acts on P^n by cyclic permutation of the factors.  ...  I would like to thank the Department of Mathematics at Michigan State University for their hospitality. I am especially grateful to Bruce Sagan for his encouragement while this project was under way.  ...

### Proofs of Conjectures about Pattern-Avoiding Linear Extensions

Colin Defant
2019 Discrete Mathematics & Theoretical Computer Science
After 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.  ...  We first consider pattern avoidance in k-ary heaps, where we obtain a general result that proves a conjecture of Levin, Pudwell, Riehl, and Sandberg in a special case.  ...  Associating complete k-ary trees with posets in the natural way, one can view a k-ary heap as a linear extension of the underlying complete k-ary tree.  ...

### Symmetric Chain Decomposition of Necklace Posets

Vivek Dhand
2012 Electronic Journal of Combinatorics
A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains.  ...  If $\mathcal{P}$ is any symmetric chain order, we prove that $\mathcal{P}^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $\mathcal{P}^n$ by cyclic permutation of the factors  ...  I would like to thank the Department of Mathematics at Michigan State University for their hospitality. I am especially grateful to Bruce Sagan for his encouragement while this project was under way.  ...

### Proofs of Conjectures about Pattern-Avoiding Linear Extensions [article]

Colin Defant
2019 arXiv   pre-print
After 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.  ...  We first consider pattern avoidance in k-ary heaps, where we obtain a general result that proves a conjecture of Levin, Pudwell, Riehl, and Sandberg in a special case.  ...  Associating complete k-ary trees with posets in the natural way, one can view a k-ary heap as a linear extension of the underlying complete k-ary tree.  ...

### On natural join of posets properties and first applications [article]

A. Krzysztof Kwaśniewski
2009 arXiv   pre-print
In addition to the three standard operations on posets which are dual of poset or ordinal and cardinal sums of partial ordered sets one adds the natural join of posets.  ...  with posing some questions arising on the way.  ...  and reliable Teacher before she as Independent Person was fired by local Bialystok University local authorities exactly on the day she had defended Rota and cobweb posets related dissertation with distinction  ...

### On polymorphism-homogeneous relational structures and their clones

Christian Pech, Maja Pech
2014 Algebra Universalis
Eventually, we completely characterize the countable polymorphism-homogeneous graphs, the polymorphism-homogeneous posets of arbitrary size, and the countable polymorphism-homogeneous strict posets.  ...  polymorphism of the structure.  ...  Let A be a countable set and let L be a meet-complete sublattice of E(A). Then L is polymorphism-homogeneous if and only if it is arithmetical.  ...

### The Complexity of the Extendibility Problem for Finite Posets

Benoit Larose, László Zádori
2003 SIAM Journal on Discrete Mathematics
In other words, if a poset admits a near unanimity operation, it also admits a totally symmetric idempotent operation of any arity. In [Fund.  ...  We generalize Pratt and Tiuryn's result on crowns by proving that EXT(P ), is NP-complete for any finite poset P which admits no nontrivial idempotent Malcev condition.  ...  The authors would like to thank Pavol Hell for patiently mediating between the referee of the paper and the authors.  ...

### Pattern occurrences in k-ary words revisited: a few new and old observations [article]

Toufik Mansour, Reza Rastegar
2022 arXiv   pre-print
In this paper, we study the pattern occurrence in k-ary words. We prove an explicit upper bound on the number of k-ary words avoiding any given pattern using a random walk argument.  ...  A simple consequence of this connection is that Wilf-equivalence of two patterns in words implies their Wilf-equivalence in permutations.  ...  We also would to thank Zachary Hunter for pointing out a gap in the proof of Theorem 1.1 in an earlier draft of this paper.  ...

### Algebraic and graph-theoretic properties of infiniten-posets

Zoltán Ésik, Zoltán L. Németh
2005 RAIRO - Theoretical Informatics and Applications
The collection of all Σ-labeled n-posets is naturally equipped with n binary product operations and n ω-ary product operations.  ...  We show that those Σ-labeled n-posets that can be generated from the singletons by the binary and ω-ary product operations form the free algebra on Σ in a variety axiomatizable by an infinite collection  ...  Poset, n-poset, composition, free algebra, equational logic. automata on ω-words, Perrin and Pin [15, 16] proposed to use free ω-semigroups, which are a two-sorted generalization of semigroups and possess  ...

### On a quasi-ordering on Boolean functions

Miguel Couceiro, Maurice Pouzet
2008 Theoretical Computer Science
Galois theory for minors of finite functions, Discrete Mathematics 254 (2002) 405-419], coincide with initial segments of the quasi-ordered set (Ω , ≤) made of the set Ω of Boolean functions, suitably  ...  Looking at examples of finitely definable classes, we show that the classes of Boolean functions with a bounded number of essential variables are finitely definable.  ...  Acknowledgments The authors would like to thank Arto Salomaa for sending a copy of the paper  , which provided the optimal lower bound given in (2) of Lemma 4.  ...

### Two-element structures modulo primitive positive constructability [article]

Manuel Bodirsky, Albert Vucaj
2020 arXiv   pre-print
In this paper, we give a complete description of the restriction P_Boole of P_fin to relational structures on a two-element set; in particular, we prove that P_Boole is a lattice.  ...  Let P_fin be the poset which arises from ordering all finite relational structures by pp-constructability. This poset is infinite, but we do not know whether it is uncountable.  ...  In other words, we require that there is exactly one occurrence of a function symbol on both sides of the equality. The use of nested terms is forbidden.  ...
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