Abstract
In this chapter we count sequences and sharings, collections and compositions, furnishing many applications and examples. Factorials and binomial coefficients are, on the one hand, indispensable tools for such counting problems, and, on the other hand, their combinatorial interpretation gives a valuable contribution in suggesting and proving many useful identities both concerning sums or alternating sums of binomials and their products.
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Notes
- 1.
James Stirling (1692–1770). We Venetians can hardly neglect to mention that he was also known as “the Venetian” after his sojourn in Venice from 1715 to 1724. Stirling had to flee from Venice with his life at risk after being accused of having stolen the secret method used for the industrial production of Murano glass.
- 2.
Here one uses the fact that if \(a_n\) and \(b_n\) are two sequences diverging to \(+\infty \) with \(a_n\sim b_n\) (\(\sim \) stands for “asymptotic to”) for \(n\rightarrow +\infty \), then \(\log _{10}a_n\sim \log _{10}b_n\). One should bear in mind that in general for a function f if \(a_n\sim b_n\) it does not necessarily follow that \(f(a_n)\sim f(b_n)\) for \(n\rightarrow +\infty \).
- 3.
Michael Stifel (1487–1567).
- 4.
Niccolò Tartaglia (1499–1557).
- 5.
Blaise Pascal (1623–1662).
- 6.
Actually, this triangle was known around the year 1000 by Indian, Persian and Chinese mathematicians.
- 7.
György Pólya (1887–1985).
- 8.
A riffle shuffle is obtained by holding one deck in each hand with the thumbs inward, then releasing the cards by the thumbs so that they fall to the table interleaved; alternatively the two decks can be put on the table and gently pushed one into the other.
- 9.
See https://discretecalculus.wordpress.com for more details on the nice periodicity properties of a Gilbreath permutation, and the explanation of more complicated tricks.
- 10.
Alexandre-Théophile Vandermonde (1735–1796).
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Mariconda, C., Tonolo, A. (2016). Counting Sequences and Collections. In: Discrete Calculus. UNITEXT(), vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-03038-8_2
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DOI: https://doi.org/10.1007/978-3-319-03038-8_2
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