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Let $S_n$ be the set of all permutations of integers from $1$ to $n$. Let $P_1$ and $P_2$ be two partitions defined on $S_n$ as follows.

$P_1$ is the set of all those permutations which have even number of cycles. $P_2$ is the set of all those permutations which have odd number of cycles. The sets $P_1$ and $P_2$ have a lexicographic ordering imposed on them.

Given a permutation $P$

  1. Tell which partition it belongs to
  2. What is the lexicographic rank($1$ based indexing) of this permutation in that partition. If this rank is $K$, you have to specify the number $K$ mod $M$ where $M = 1000000007 (1e9 + 7)$

Here $N$ can be as large as $10^5$

My approach: Calculate the number of cycles in P using a simple DFS technique on my computer and decide which partition it belongs to. For calculating rank I first calculate its rank in the set $S_n$ Let that rank be $K_1$($1$ based indexing) then the actual rank is coming out to be floor($\frac{K_1+1}{2}$). I am having trouble applying modular arithmetic on this expression. Is there any other way to solve this problem?

Note: By "cycle" I mean when the permutation is represented as a graph, the number of components or disjoint forests formed.

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  • $\begingroup$ @AndrásSalamon I have corrected the question $\endgroup$ – lassaendie Oct 5 '15 at 11:41
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I assume by "even length cycles" and "odd length cycles" you mean the Parity/Sign of a permutation ( https://en.wikipedia.org/wiki/Parity_of_a_permutation ). In this case you might use one of the following methods to compute it: https://mathoverflow.net/questions/72669/finding-the-parity-of-a-permutation-in-little-space

https://math.stackexchange.com/questions/65923/how-does-one-compute-the-sign-of-a-permutation

For the other question: Determining the lexicographic ordering of the permutation you could use the Factorial Number system and the Lehmer-Code: https://en.wikipedia.org/wiki/Factorial_number_system#Permutations

See for example : http://orgesleka.blogspot.de/2015/09/candidate-one-way-function.html for a Python-Implementation of the Lehmer-Code and related Factorial-Number System.

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