## COMBINATORIAL PROPERTIES OF PERMUTATION TABLEAUX

### BibTeX

@MISC{Burstein_combinatorialproperties,

author = {Alexander Burstein and Niklas Eriksen},

title = {COMBINATORIAL PROPERTIES OF PERMUTATION TABLEAUX},

year = {}

}

### OpenURL

### Abstract

Abstract. Abstract. We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between 1-hinge and 0-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that are known to be equidistributed via a complicated map to permutations that translates those to occurrences of certain patterns. We give two direct maps on tableaux that proves the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux. 1.

### Citations

376 |
The On-Line Encyclopedia of Integer Sequences,” http://www.research.att.com/∼njas/sequences/index.html
- Sloane
(Show Context)
Citation Context ...< 0 or k > (n − 2)/2 for n ≥ 2. Routine application of generating functions now yields the theorem. □ Corollary 5.5. Proof. Let g(n) = |Mn(123, 213)|. Then g(n) = |Mn(123, 213)| = A002965(n + 1) (see =-=[9]-=-), n ≥ 0. ∑⌊ n−2 2 ⌋ k=0 f(n, k), so summing Equation (5.1) over k, we get g(n) = 2g(n − 2) + g(n − 4), n ≥ 4, and g(0) = 1, g(1) = 1, g(2) = 1, g(3) = 2, which implies the first equality in the theor... |

77 |
The On-Line Encyclopedia of Integer Sequences, 2000, published electronically at http://www.research.att.com/ ~ njas/sequences
- Sloane
(Show Context)
Citation Context ... of permutations in Mn(123, 213) with k nonessential 1s, then |M k 2n(123, 213)| = a(2n − 2, k) + a(2n − 3, k), |M k 2n+1(123, 213)| = 2a(2n − 2, k) + a(2n − 3, k), where a(n, k) = A037027(n, k) (see =-=[11]-=-), the kth entry in row n of the Fibonacci-Pascal triangle. Proof: Let π ∈ M k n(123, 213) and let T = Φ −1 (π) be its permutation tableau. Simply avoidance of the both pattern implies the following: ... |

48 | Enumeration of Totally Positive Grassmann Cells - Williams |

42 |
Permutation tableaux and permutation patterns
- Steingrimsson, Williams
(Show Context)
Citation Context ... are in bijection with permutations. They originally turned up in the enumeration of totally positive Grassmannian cells [8, 11]. Permutation tableaux have then been studied either in their own right =-=[1, 10]-=- to produce enumeration results for permutations, or in connection with the PASEP model in statistical mechanics [3, 5, 6]. A permutation tableaux T is usually defined as a k × (n − k) array filled wi... |

36 | Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials
- Corteel, Williams
(Show Context)
Citation Context ... [8, 11]. Permutation tableaux have then been studied either in their own right [1, 10] to produce enumeration results for permutations, or in connection with the PASEP model in statistical mechanics =-=[3, 5, 6]-=-. A permutation tableaux T is usually defined as a k × (n − k) array filled with zeroes, ones and twos such that the cells filled with zeroes and ones form a Young tableau Yλ of an integer partition λ... |

24 | On some properties of permutation tableaux
- Burstein
- 2006
(Show Context)
Citation Context ... are in bijection with permutations. They originally turned up in the enumeration of totally positive Grassmannian cells [8, 11]. Permutation tableaux have then been studied either in their own right =-=[1, 10]-=- to produce enumeration results for permutations, or in connection with the PASEP model in statistical mechanics [3, 5, 6]. A permutation tableaux T is usually defined as a k × (n − k) array filled wi... |

18 |
Crossings and alignments of permutations
- Corteel
(Show Context)
Citation Context ... [8, 11]. Permutation tableaux have then been studied either in their own right [1, 10] to produce enumeration results for permutations, or in connection with the PASEP model in statistical mechanics =-=[3, 5, 6]-=-. A permutation tableaux T is usually defined as a k × (n − k) array filled with zeroes, ones and twos such that the cells filled with zeroes and ones form a Young tableau Yλ of an integer partition λ... |

16 | A Markov chain on permutations which projects to the PASEP
- Corteel, Williams
- 2007
(Show Context)
Citation Context ... [8, 11]. Permutation tableaux have then been studied either in their own right [1, 10] to produce enumeration results for permutations, or in connection with the PASEP model in statistical mechanics =-=[3, 5, 6]-=-. A permutation tableaux T is usually defined as a k × (n − k) array filled with zeroes, ones and twos such that the cells filled with zeroes and ones form a Young tableau Yλ of an integer partition λ... |

10 | Enumeration of totally positive Grassmann cells, Adv. in Math. 190 (2005), 319–342. (Jean-Christophe Aval) LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, FRANCE E-mail address: aval@labri.fr URL: http://www.labri.fr/perso/a - Williams |

9 |
Strongly Convergent Games and Coxeter Groups
- Eriksson
- 1993
(Show Context)
Citation Context ...y changes induced by one of the cells will not affect the paths through another one of these cells, and hence will not affect the changes induced by that other cell. By the strong convergence theorem =-=[7]-=-, it suffices to show that any two moves that are valid at the same time commute and that the sequence of moves is finite for a game to have a unique end result. We have shown that any two moves commu... |

3 |
A conjecture of Stanley on alternating permutations, Electron
- Chapman, Williams
(Show Context)
Citation Context ...considering pattern-restricted sets when the information about descents and inversions can be translated into information about weak excedances and deficiencies. A good example of such a situation is =-=[2]-=- where alternating permutations (descent-related objects) with the maximum number of fixed points (excedance-related property) are considered. In a 1-hinge tableau T , those ones that are not forced b... |

2 | A simple bijection between permutation tableaux and permutations
- Corteel
(Show Context)
Citation Context ...edances in π and conclude that wex(π) = k. We also note that a bijection between permutations tableaux and permutations with n − k descents (elements π(i) such that π(i) > π(i + 1)) has been proposed =-=[4]-=-, which amongst other things reproves that descents and deficiencies are equidistributed. The inverse Γ = Φ −1 : Sn → Tn of the presented bijection is less natural, and several algorithms describing Γ... |

1 |
Webs in totally positive Grassmann cells, manuscript
- Postnikov
- 2001
(Show Context)
Citation Context ...n tableaux. 1. Introduction Permutation tableaux are combinatorial objects that are in bijection with permutations. They originally turned up in the enumeration of totally positive Grassmannian cells =-=[8, 11]-=-. Permutation tableaux have then been studied either in their own right [1, 10] to produce enumeration results for permutations, or in connection with the PASEP model in statistical mechanics [3, 5, 6... |