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Séminaire Lotharingien de Combinatoire, 86B.1 (2022), 12 pp.

# Andrew Elvey Price

# Enumeration of Walks with Small Steps Avoiding a Quadrant

**Abstract.**
We address the enumeration of walks with weighted small steps avoiding
a quadrant. In particular we give an exact, integral-expression
solution for the generating function **C**(*x*,*y*;*t*) counting these
walks by length and end-point. Moreover, we determine precisely when
this generating function is algebraic, D-finite or D-algebraic with
respect to *x*, showing that this complexity is the same as for walks
in the quarter-plane with the same starting point, as long as the
starting point (*p*,*q*) of the walks lies in the quarter plane
then. Finally, we give an integral-free expression for the solution in
the cases where (*p*,*q*) lies just outside the quarter plane, that is
*p*=0 or *q*=0 with our convention, proving a conjecture of Raschel
and Trotignon.

Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.

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