Sign-Balanced Posets

✍ Scribed by Dennis E. White

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 345 KB
• Volume: 95
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2000.3146

✦ Synopsis

Let P be a finite partially ordered set with a fixed labeling. The sign of a linear extension of P is its sign when viewed as a permutation of the labels of the elements of P. Call P sign-balanced if the number of linear extensions of P of positive sign is the same as the number of linear extensions of P of negative sign. In this paper we determine when the posets in a particular class are sign-balanced. When posets in this class are not sign-balanced, we determine the difference between the number of positive linear extensions and the number of negative linear extensions. One special case of this class is the product of an m-chain with an n-chain, m and n both >1. In this case, we show P is sign-balanced if and only if m#n mod 2.

2001 Academic Press

? # L f (P) q inv(?) .

Bjo rner and Wachs [1] have shown that if the Hasse diagram of P is a tree and f is a postorder labeling, then INV P, f (q) can be written as a product