### Abstract

In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs provide

many new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we discuss several bijective questions linking our patterns to other combinatorial objects.

Original language | English |
---|---|

Article number | A11 |

Pages (from-to) | 129-154 |

Number of pages | 26 |

Journal | Integers: Electronic Journal of Combinatorial Number Theory |

Volume | 10 |

Publication status | Published - 2010 |

### Fingerprint

### Keywords

- place-difference-value patterns
- word patterns
- permutations
- permutation patterns theory

### Cite this

*Integers: Electronic Journal of Combinatorial Number Theory*,

*10*, 129-154. [A11].

}

*Integers: Electronic Journal of Combinatorial Number Theory*, vol. 10, A11, pp. 129-154.

**Place-difference-value patterns : a generalization of generalized permutation and word patterns.** / Kitaev, Sergey; Remmel, Jeffrey.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Place-difference-value patterns

T2 - a generalization of generalized permutation and word patterns

AU - Kitaev, Sergey

AU - Remmel, Jeffrey

PY - 2010

Y1 - 2010

N2 - Motivated by the study of Mahonian statistics, in 2000, Babson and Steingr´ımsson introduced the notion of a “generalized permutation pattern” (GP) which generalizes the concept of “classical” permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingr´ımsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-M´elou et al. introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets.In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs providemany new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we discuss several bijective questions linking our patterns to other combinatorial objects.

AB - Motivated by the study of Mahonian statistics, in 2000, Babson and Steingr´ımsson introduced the notion of a “generalized permutation pattern” (GP) which generalizes the concept of “classical” permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingr´ımsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-M´elou et al. introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets.In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs providemany new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we discuss several bijective questions linking our patterns to other combinatorial objects.

KW - place-difference-value patterns

KW - word patterns

KW - permutations

KW - permutation patterns theory

UR - http://www.integers-ejcnt.org/vol10.html

M3 - Article

VL - 10

SP - 129

EP - 154

JO - Integers: Electronic Journal of Combinatorial Number Theory

JF - Integers: Electronic Journal of Combinatorial Number Theory

SN - 1553-1732

M1 - A11

ER -