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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 inclusionexclusion (and, in fact, it is one of the … Continue reading
Posted in combinatorics, derangements, permutations, recurrence relations
5 Comments
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
Posted in combinatorics
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Parametric Derivatives
What is the th derivative of a parametricallydefined function? For the th derivative of the product of two functions there is Leibniz’s rule, and for the th derivative of the composition of two functions there is Faà di Bruno’s formula. … Continue reading
Posted in calculus, special functions
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Binomial Coefficient Identities via Integration
As the title indicates, this post is going to look at proving some binomial coefficient identities via integration. The first set of identities starts with the expression Now, evaluate this expression in two ways: Do the integration, Swap the sum … Continue reading
Posted in binomial coefficients, calculus
2 Comments