PUMaC 2008 · 组合（A 组） · 第 10 题

PUMaC 2008 — Combinatorics (Division A) — Problem 10

(7 points) In his youth, Professor John Horton Conway lived on a farm with 2009 cows. Conway wishes to move the cows from the negative x axis to the positive x axis. The cows are initially lined up in order 1 , 2 , . . . , 2009 on the negative x axis. Conway can give two possible commands to the cows. One is the PUSH command, upon which the first cow from the negative x axis moves to the lowest position on the positive y axis. The other is the POP command, upon which the cow in the lowest position on the y axis moves to the positive x axis. For example, if Conway says PUSH POP 2009 times, then the resulting permutation of cows is the same, 1 , 2 , . . . , 2009. If Conway says PUSH 2009 times followed by POP 2009 times, the resulting permutation of cows is 2009 , . . . , 2 , 1. How many output permutations are possible after Conway finishes moving all the cows from the negative x axis to the positive x axis? 2

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