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Séminaire Lotharingien de Combinatoire, B70j (2014), 50 pp.

# Maciej Dołęga, Valentin Féray and Piotr Śniady

# Jack Polynomials and Orientability Generating Series of Maps

**Abstract.**
We study *Jack characters*,
which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization.
These quantities have been introduced by Lassalle who formulated some challenging conjectures about them.
We conjecture the existence of a weight on non-oriented
maps (i.e., graphs drawn on non-oriented surfaces)
which allows to express
any given Jack character as a weighted sum of some simple functions indexed by maps.
We provide a candidate for this weight which gives a positive answer to our conjecture in some,
but unfortunately not all, cases.
In particular, it gives a positive answer for Jack characters specialized
to Young diagrams of rectangular shape.
This candidate weight attempts to measure,
in a sense, the non-orientability of a given map.

Received: April 25, 2013.
Revised: September 15, 2014.
Accepted: October 15, 2014.

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