# Monthly Archives: November 2011

## Another Nice Proof of the Derangements Formula

Strangely enough, a couple of days after I made my last post on a way to prove the derangements formula that I had never seen before, I came across another new (to me) way to prove the derangements formula in … Continue reading

## A Nice Proof of the Derangements Formula

A derangement is a permutation in which none of the elements is mapped to itself.  The formula for the number of derangements on elements is This formula is normally proved using inclusion-exclusion (and, in fact, it is one of the … Continue reading

## A Product Calculus

The derivative can be thought of as a measure of how much changes relative to changes in .  But why measure the relationship between the rates of change this way?  Why not a different definition of the derivative that compares … Continue reading

Posted in calculus | 2 Comments

## Permutations with Fixed Points and Successions

Suppose we have a permutation on the set .  A fixed point of is a value such that .  If , then we say has a succession at .  Take, for example, the permutation 213596784.  It has four fixed points: … Continue reading

Posted in combinatorics, permutations | 3 Comments

## Euler Numbers via Eulerian Numbers

This is a little curiosity I discovered several years ago. The Euler numbers are the coefficients in the Maclaurin expansion of hyperbolic secant: The Eulerian numbers count the number of permutations on elements with ascents; i.e., the number of permutations … Continue reading