scholarly journals Generating Trees and Other Combinatorial Objects Lexicographically

1979 ◽  
Vol 8 (1) ◽  
pp. 73-81 ◽  
Author(s):  
S. Zaks ◽  
D. Richards
Keyword(s):  
Combinatorial Objects ◽  
Generating Trees

scholarly journals Ascent Sequences Avoiding Pairs of Patterns

10.37236/4479 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Andrew M. Baxter ◽  
Lara K. Pudwell
Keyword(s):  
Pattern Avoidance ◽  
Exact Enumeration ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
Generating Trees ◽  
Precise Bound ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations

Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrímsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdős-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound.

scholarly journals Designing locally maximally entangled quantum states with arbitrary local symmetries

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 450
Author(s):  
Oskar Słowik ◽  
Adam Sawicki ◽  
Tomasz Maciążek
Keyword(s):  
Quantum State ◽  
Quantum States ◽  
Local Mode ◽  
Trivial Representation ◽  
Tensor Power ◽  
Technical Result ◽  
Combinatorial Objects ◽  
Local Symmetries ◽  
Entangled Quantum States ◽  
Group Of Symmetries

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the Nth tensor power of any irreducible representation of SU(N) contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.

scholarly journals Corners in Tree-Like Tableaux

10.37236/5712 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Paweł Hitczenko ◽  
Amanda Lohss
Keyword(s):  
Exclusion Process ◽  
Limiting Distribution ◽  
Tree Structure ◽  
Random Tree ◽  
Type B ◽  
Combinatorial Objects ◽  
Simple Exclusion Process ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
Symmetric Tree

In this paper, we study tree–like tableaux, combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree–like tableaux and the total number of corners in symmetric tree–like tableaux. In this paper, we prove both conjectures. Our proofs are based off of the bijection with permutation tableaux or type–B permutation tableaux and consequently, we also prove results for these tableaux. In addition, we derive the limiting distribution of the number of occupied corners in random tree–like tableaux and random symmetric tree–like tableaux.

A SURVEY OF FACTORIZATION COUNTING FUNCTIONS

2005 ◽  
Vol 01 (04) ◽  
pp. 563-581 ◽  
Author(s):  
A. KNOPFMACHER ◽  
M. E. MAYS
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Additive Number Theory ◽  
Recursive Algorithms ◽  
Additive Number ◽  
Combinatorial Objects ◽  
Survey Techniques ◽  
Counting Functions ◽  
General Field ◽  
Positive Integers

The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to ±1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a product of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.

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