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Monthly Archives: December 2011
The Number of NSEW Lattice Paths with a Maximum Height of y
I’ve been thinking and learning some about lattice path statistics lately, and a couple of days ago I worked out the answer to the following question (with help from some suggestions from Anthony Quas via Math Overflow). Suppose we have … Continue reading
Posted in combinatorics, lattice paths
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Checking Multiplication via Digit Sums
Last week a friend who is a fourth grade teacher came to me with a math problem. The father of one of his students had showed him a trick for checking the result of a threedigit multiplication problem. The father … Continue reading
Posted in arithmetic, elementary number theory
4 Comments
A Combinatorial Proof of a Formula Involving Euler’s Totient Function
Several years ago I found a proof of the following identity, which I had not seen before (and still haven’t seen anywhere else): where is Euler’s totient function, the number of positive integers less than or equal to and relatively … Continue reading
Posted in combinatorics, elementary number theory
2 Comments
The Duality “Gap” in Linear Programming
Wait… what? Doesn’t the strong duality theorem imply that there is no duality gap in linear programming? Mostly, yes, although there is a sense in which there actually is a duality gap; it occurs when the primal and dual problems … Continue reading
Posted in linear programming, optimization
3 Comments
Solving TwoTerm Combinatorial Recurrence Relations
Many named combinatorial numbers satisfy the following general twoterm combinatorial recurrence relation: For example: Binomial coefficients: Stirling cycle numbers: Stirling subset numbers: Lah numbers: Eulerian numbers: Secondorder Eulerian numbers: It is not known how to solve the recurrence in general. … Continue reading
Posted in combinatorics, recurrence relations
1 Comment
Harmonic Numbers and Stirling Numbers
This post will present three proofs of the following representation of harmonic numbers in terms of Stirling cycle numbers (or unsigned Stirling numbers of the first kind): Each proof uses a different property of the Stirling numbers. Proof 1: … Continue reading
Posted in combinatorics, sequences and series
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Generating Geometric and Truncated Exponential Random Variates
If you do numerical simulations you often find yourself needing to generate random variates from a specific distribution. There are several techniques for doing this, such as the inverse transform method and the acceptanceandrejection method. This post will talk about … Continue reading
Posted in probability, simulation
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Using Indicator Variables to Calculate Expected Values
Indicator variables can often be used to find an expected value when doing the calculation directly from the definition would be quite difficult. This post will look at a couple of examples and give references to several more. First example … Continue reading
Posted in probability
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