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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Parking functions, stack-sortable permutations, and spaces of paths in the Johnson graph | Parking functions stack-sortable permutations and spaces of paths in the Johnson graph Catalin Zara Department of Mathematics Yale University New Haven CT USA zara@math.yale.edu Submitted Apr 4 2002 Accepted Apr 10 2003 Published Apr 23 2003 MR Subject Classifications 05A05 05C38 Abstract We prove that the space of possible final configurations for a parking problem is parameterized by the vertices of a regular Bruhat graph associated to a 231-avoiding permutation and we show how this relates to parameterizing certain spaces of paths in the Johnson graph. 1 Introduction There are n seats on a row in a cinema hall numbered 1 . n starting from the aisle n k seats are already taken but seats q q1 q2 qk are still available. Some k late patrons want to take these seats they enter the row and start advancing. When they are in front of seats p p-1 p2 pk the lights go off. Will they be able to find the empty seats They can only advance towards the end of the row but not go back. Also a patron can pass a patron in front of him only if that second patron is already seated. Intuitively it is like the late patrons are choosing their seats in the order in which they entered the row. It is clear that this is impossible if for some i the patron in front of seat pi for short patron pi passed more than k 1 i empty seats so we assume that pi qi for all i 1 . k if this condition is satisfied we say that p N q and call p q an initial position. A possible final arrangement is given by a permutation u E Sk for each i 1 . k patron pi takes the seat qu i . For an initial position p q not all permutations might appear since a patron can t go back we call a permutation u E Sk attainable from p q or simply attainable when no confusion is possible if u can appear as the permutation corresponding to a final arrangement if the initial position is p q. For example if everyone knows how many empty seats he passed by then for each i 1 . k the patron pi could take the ith available seat qi the .
