The Umbral Transfer-Matrix Method. I. Foundations

According to cohen venavi, a classic Hebrew essay by Ekhad Ha-Am, there are two ways of serving God: Priest and Prophet. The priest is a very skilled technician who knows by heart the 613 mitzvahs, and who can slaughter a lamb with due regard to the many rigorous and subtle laws. On the other hand, a prophet is often very clumsy in the day-to-day ritual, and is unable to sacrifice an animal properly. But prophets have a direct line to God. In the long run, their impact and influence far surpass those of the priests. Do we not all know the names of Isaiah, Jeremiah, and Ejekiel? Yet none of us remembers any of the names of the high priests who lived at the same time. Gian-Carlo Rota was a paragon of the mathematical prophet. His technical contributions, while substantial, dwarf in comparison to his vision, insight, and new unifying concepts. Also, his uncanny realizations that some things are important have revolutionized more than one combinatorial area. Rota's dislike of routine priestly work is expressed nicely in the following extract from Richard Stanley's touching and warm obituary that appeared in SIAM News: doi:10.1006 jcta.2000.3110, available online at http: www.idealibrary.com on