Question About HW 1, Problem 5

Is the expression on the right-hand side of the equation correct? Here’s why I’m asking: For k = 2, the denominator of this expression is 1 - 2x + x^2, or (1 - x)^2, so the fraction reduces to 1/(1 - x). But this implies that c_2 (n) = 1 for all n, which is not true.

 

David

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3 comments

  1. Maybe it is supposed to say that every part is strictly less than k, not less than or equal to. Because then that would be true since it’s just the one composition of all 1’s, right?

    Hannah

  2. Hannah is right; c_k(n) should be the number of compositions of n such that all parts are *strictly less* than k. Hannah showed us how to check the case k=2, and I encourage you to also think about the case k=3.
    Thanks for the heads up, David.

  3. Frits Scholer

    EC1 problem 26 uses (k+1) so that would be the correct exponent in the formula. Love your lectures after buying the book (spent 2 days on hw1 problem 3 and found alternative solution based on induction) Very stimulating problems.
    Frits, Amsterdam

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