Enumeration of derangements with descents in prescribed positions

We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated

新奇層状超伝導体に対する第一原理フォノン計算

Supervisor:前園　涼先端科学技術研究科博

Sorting a bridge hand www.elsevier.com/locate/disc

Sorting a permutation by block moves is a task that every bridge player has to solve every time she picks up a new hand of cards. It is also a problem for the computational biologist, for block moves are a fundamental type of mutation that can explain why genes common to two species do not occur in the same order in the chromosome. It is not known whether there exists an optimal sorting procedure running in polynomial time. Bafna and Pevzner gave a polynomial time algorithm that sorts any permutation of length n in at most 3n=4 moves. Our new algorithm improves this to ⌊(2n−2)=3 ⌋ for n ¿ 9. For the reverse permutation, we give an exact expression for the number of moves needed, namely ⌈(n +1)=2⌉. Computations of Bafna and Pevzner up to n = 10 seemed to suggest that this is the worst case; but as it turns out, a rst counterexample occurs for n = 13, i.e. the bridge player’s case. Professional card players never sort by rank, only by suit. For this case, we give a complet