Rationality, irrationality, and Wilf equivalence in generalized factor order

Sergey Kitaev, Jeff Liese, Jeffrey Remmel, Bruce Sagan

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let P be a partially ordered set and consider the free monoid P∗ of all words over P. If w,w′∈P∗ then w′ is a factor of w if there are words u,v with w=uw′v. Define generalized factor order on P∗ by letting u≤w if there is a factor w′ of w having the same length as u such that u≤w′, where the comparison of u and w′ is done component wise using the partial order in P. One obtains ordinary factor order by insisting that u=w′ or, equivalently, by taking P to be an antichain.

Given u∈P∗, we prove that the language F(u)={w : w≥u} is accepted by a finite state automaton. If P is finite then it follows that the generating function F(u)=∑w≥uw is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order.

We also consider P=P, the positive integers with the usual total order, so that P∗ is the set of compositions. In this case one obtains a weight generating function F(u;t,x) by substituting txn each time n∈P appears in F(u). We show that this generating function is also rational by using the transfer-matrix method. Words u,v are said to be Wilf equivalent if F(u;t,x)=F(v;t,x) and we prove various Wilf equivalences combinatorially.

Björner found a recursive formula for the Möbius function of ordinary factor order on P∗. It follows that one always has μ(u,w)=0,±1. Using the Pumping Lemma we show that the generating function M(u)=∑w≥u|μ(u,w)|w can be irrational.
Original languageEnglish
Article numberR22
Number of pages26
JournalThe Electronic Journal of Combinatorics
Volume16
Issue number2
Publication statusPublished - 2009

Fingerprint

Irrationality
Rationality
Equivalence
Generating Function
Free Monoid
Subword
Finite State Automata
Antichain
Transfer matrix method
Transfer Matrix Method
Recursive Formula
Partially Ordered Set
Finite automata
Partial Order
Weight Function
Lemma
Analogue
Integer
Chemical analysis
Theorem

Keywords

  • factor order
  • word order
  • weight generating function

Cite this

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title = "Rationality, irrationality, and Wilf equivalence in generalized factor order",
abstract = "Let P be a partially ordered set and consider the free monoid P∗ of all words over P. If w,w′∈P∗ then w′ is a factor of w if there are words u,v with w=uw′v. Define generalized factor order on P∗ by letting u≤w if there is a factor w′ of w having the same length as u such that u≤w′, where the comparison of u and w′ is done component wise using the partial order in P. One obtains ordinary factor order by insisting that u=w′ or, equivalently, by taking P to be an antichain.Given u∈P∗, we prove that the language F(u)={w : w≥u} is accepted by a finite state automaton. If P is finite then it follows that the generating function F(u)=∑w≥uw is rational. This is an analogue of a theorem of Bj{\"o}rner and Sagan for generalized subword order.We also consider P=P, the positive integers with the usual total order, so that P∗ is the set of compositions. In this case one obtains a weight generating function F(u;t,x) by substituting txn each time n∈P appears in F(u). We show that this generating function is also rational by using the transfer-matrix method. Words u,v are said to be Wilf equivalent if F(u;t,x)=F(v;t,x) and we prove various Wilf equivalences combinatorially.Bj{\"o}rner found a recursive formula for the M{\"o}bius function of ordinary factor order on P∗. It follows that one always has μ(u,w)=0,±1. Using the Pumping Lemma we show that the generating function M(u)=∑w≥u|μ(u,w)|w can be irrational.",
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year = "2009",
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Rationality, irrationality, and Wilf equivalence in generalized factor order. / Kitaev, Sergey; Liese, Jeff; Remmel, Jeffrey; Sagan, Bruce.

In: The Electronic Journal of Combinatorics, Vol. 16, No. 2, R22, 2009.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Rationality, irrationality, and Wilf equivalence in generalized factor order

AU - Kitaev, Sergey

AU - Liese, Jeff

AU - Remmel, Jeffrey

AU - Sagan, Bruce

PY - 2009

Y1 - 2009

N2 - Let P be a partially ordered set and consider the free monoid P∗ of all words over P. If w,w′∈P∗ then w′ is a factor of w if there are words u,v with w=uw′v. Define generalized factor order on P∗ by letting u≤w if there is a factor w′ of w having the same length as u such that u≤w′, where the comparison of u and w′ is done component wise using the partial order in P. One obtains ordinary factor order by insisting that u=w′ or, equivalently, by taking P to be an antichain.Given u∈P∗, we prove that the language F(u)={w : w≥u} is accepted by a finite state automaton. If P is finite then it follows that the generating function F(u)=∑w≥uw is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order.We also consider P=P, the positive integers with the usual total order, so that P∗ is the set of compositions. In this case one obtains a weight generating function F(u;t,x) by substituting txn each time n∈P appears in F(u). We show that this generating function is also rational by using the transfer-matrix method. Words u,v are said to be Wilf equivalent if F(u;t,x)=F(v;t,x) and we prove various Wilf equivalences combinatorially.Björner found a recursive formula for the Möbius function of ordinary factor order on P∗. It follows that one always has μ(u,w)=0,±1. Using the Pumping Lemma we show that the generating function M(u)=∑w≥u|μ(u,w)|w can be irrational.

AB - Let P be a partially ordered set and consider the free monoid P∗ of all words over P. If w,w′∈P∗ then w′ is a factor of w if there are words u,v with w=uw′v. Define generalized factor order on P∗ by letting u≤w if there is a factor w′ of w having the same length as u such that u≤w′, where the comparison of u and w′ is done component wise using the partial order in P. One obtains ordinary factor order by insisting that u=w′ or, equivalently, by taking P to be an antichain.Given u∈P∗, we prove that the language F(u)={w : w≥u} is accepted by a finite state automaton. If P is finite then it follows that the generating function F(u)=∑w≥uw is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order.We also consider P=P, the positive integers with the usual total order, so that P∗ is the set of compositions. In this case one obtains a weight generating function F(u;t,x) by substituting txn each time n∈P appears in F(u). We show that this generating function is also rational by using the transfer-matrix method. Words u,v are said to be Wilf equivalent if F(u;t,x)=F(v;t,x) and we prove various Wilf equivalences combinatorially.Björner found a recursive formula for the Möbius function of ordinary factor order on P∗. It follows that one always has μ(u,w)=0,±1. Using the Pumping Lemma we show that the generating function M(u)=∑w≥u|μ(u,w)|w can be irrational.

KW - factor order

KW - word order

KW - weight generating function

UR - http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i2r22

M3 - Article

VL - 16

JO - The Electronic Journal of Combinatorics

JF - The Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

M1 - R22

ER -