Download Algebraic Combinatorics and Computer Science: A Tribute to by H. Crapo (auth.), H. Crapo, D. Senato (eds.) PDF

By H. Crapo (auth.), H. Crapo, D. Senato (eds.)

This ebook, devoted to the reminiscence of Gian-Carlo Rota, is the results of a collaborative attempt by means of his neighbors, scholars and admirers. Rota was once one of many nice thinkers of our instances, innovator in either arithmetic and phenomenology. i think moved, but touched by way of a feeling of unhappiness, in offering this quantity of labor, regardless of the terror that i'll be unworthy of the duty that befalls me. Rota, either the scientist and the guy, used to be marked via a generosity that knew no bounds. His rules opened broad the horizons of fields of analysis, allowing an astounding variety of scholars from all around the globe to develop into enthusiastically concerned. The contagious strength with which he established his super psychological capability continuously proved clean and encouraging. past his renown as proficient scientist, what was once rather awesome in Gian-Carlo Rota used to be his skill to understand the various highbrow capacities of these earlier than him and to evolve his communications consequently. This human experience, complemented through his acute appreciation of the significance of the person, acted as a catalyst in bringing forth the superior in every one of his scholars. Whosoever used to be lucky adequate to take pleasure in Gian-Carlo Rota's longstanding friendship was once so much enriched by way of the adventure, either mathematically and philosophically, and had social gathering to understand son cote de bon vivant. The e-book opens with a heartfelt piece via Henry Crapo within which he meticulously items jointly what Gian-Carlo Rota's premature loss of life has bequeathed to science.

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Additional resources for Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota

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The basic tools of combinatorial mathematics are the elementary symmetric functions, to wit, the following polynomials in n variables: + X2 + ... + Xn , + XIX3 + ... + Xn-IXn, el (XI, X2, ... ,xn ) = XI e2(xl, X2,··· ,xn ) = XIX2 en-I (XI, X2, ... ,xn ) = X2X3'" Xn + XIX3X4'" Xn + ... + XIX2'" Xn-I, en (xl,x2, ... ,xn ) =XI X2"' Xn' Observe an interesting coincidence. The last of these n symmetric functions is also the formula for the volume of a parallelotope. Axiom 4 can be rewritten as Let us try an experiment, and replace the n-th symmetric function by the (n - I )-st symmetric function.

I know of no person who has an intuitive feeling for the mean width, similar to the intuitive feeling we have for volume and area. Let us conjecture a possible application of the mean width. A potato grower knows that a potato's volume is important, because it determines the nutritional content of the potato. The potato grower also knows that the surface area of a potato is important, because it is rumored that the vitamins in a potato are concentrated in the skin. We may conjecture that, as soon as the potato grower will become aware of the mean width, he or she will find a nutritional interpretation of the mean width of a potato.

Cyclic symmetric functions, satisfying the four equations + I(XI, X4, X2, X3) + I(XI, X3, X4, X2) = a, I(XI, X2, X3, X4) + I(X4, X2, Xl, X3) + I(X3, X2, X4, xJ) = a, I(XI, X2, X3, X4) + I(X4, Xl, X3, X2) + I(X2, X4, X3, Xl) = a, I(XI, X2, X3, X4) + I(X3, Xl, X2, X4) + I(X2, X3, Xl, X4) = a. I(XI, X2, X3, X4) 4. Functions satisfying the four equations I(XI, X2, X3, X4) + I(X2, Xl, X3, X4) + I(XI, X2, X4, + I(X2, Xl, X4, X3) = I(XI, X2, X3, X4) L X3) + I(X3, X2, Xl, X4) + I(XI, X4, X3, X2) + I(X3,X4,XI,X2) = I(XI, X2, X3, X4) a, + I(XI, X3, X2, X4) + I(X4, X3, X2, Xl) sign(o-)I(xul, xu2, xu3, X( 4) a, + I(X4, X2, X3, xJ} = a, = a.

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