Abstract
The binomial distribution is the first probability distribution a student encounters. Among its many properties is the fact that it is palindromic and unimodal. Many combinatorial distributions, including the Eulerian and Narayana distributions, can be built out of copies of binomial distributions that are shifted to have the same center of symmetry, and this fact has many interesting consequences.
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Notes
- 1.
In the literature the term “symmetric” is sometimes used to describe what we mean by “palindromic.” This is okay in some circumstances, but there is a more common notion of “symmetric polynomial”—namely a polynomial that is fixed under permutation of its variables—so we prefer the less ambiguous term. George Andrews used another synonym for palindromic, “reciprocal polynomial,” in [8] and [9].
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Petersen, T.K. (2015). Gamma-nonnegativity. In: Eulerian Numbers. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-3091-3_4
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