On two combinatorial inequalities that explain the blimpy shape of heady- s and taily- s bit strings

We prove two combinatorial inequalities introduced in our prior study of the “blimpy” graphical shape of the number of bit strings with a given score under an interesting scoring system. Generating functions are used to establish the inequalities, which in turn imply two of the salient graphical features, unimodality near the zero score, and shape asymmetry for positive versus negative scores. One inequality provides a lower bound on the expected value of a discrete random variable with probabilities proportional to a product of two binomial coefficients. The other inequality states that the expected value with respect to peri-central binomial coefficients of other binomial coefficients lying on an oblique ray in Pascal’s triangle exceeds the expected value along an adjacent parallel ray to its left.