# Journal of Combinatorics

## Volume 9 (2018)

### Pattern avoiding linear extensions of rectangular posets

Pages: 185 – 220

DOI: http://dx.doi.org/10.4310/JOC.2018.v9.n1.a9

#### Authors

David Anderson (Thumbtack, Inc., San Francisco, California, U.S.A.)

Eric S. Egge (Department of Mathematics and Statistics, Carleton College, Northfield, Minnesota, U.S.A.)

Manda Riehl (Department of Mathematics, University of Wisconsin, Eau Claire, Wisc., U.S.A.)

Lucas Ryan (Department of Mathematics and Statistics, Carleton College, Northfield, Minnesota, U.S.A.)

Ruth Steinke (Department of Mathematics and Statistics, Carleton College, Northfield, Minnesota, U.S.A.)

Yuriko Vaughan (Target Corporation, Minneapolis, Minnesota, U.S.A.)

#### Abstract

Inspired by Yakoubov’s 2015 investigation of pattern avoiding linear extensions of the posets called combs, we study pattern avoiding linear extensions of rectangular posets. These linear extensions are closely related to standard tableaux. For positive integers $s$ and $t$ we consider two natural rectangular partial orders on $\lbrace 1, 2, \dotsc , st \rbrace$, which we call the NE rectangular order and the EN rectangular order. First we enumerate linear extensions of both rectangular orders avoiding most sets of patterns of length three. Then we use both a generating tree and a bijection to show that the linear extensions of the EN rectangular order which avoid $1243$ are counted by the Fuss–Catalan numbers. Next we use the transfer matrix method to enumerate linear extensions of the EN rectangular order which avoid $2143$. Finally, we open an investigation of the distribution of the inversion number on pattern avoiding linear extensions.

#### Keywords

Catalan number, lattice path, linear extension, partially ordered set, pattern avoiding permutation, standard tableaux

#### 2010 Mathematics Subject Classification

05A05, 05A15

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