By Richard P. Stanley

This moment quantity of a two-volume easy creation to enumerative combinatorics covers the composition of producing features, timber, algebraic producing capabilities, D-finite producing features, noncommutative producing services, and symmetric features. The bankruptcy on symmetric services offers the single to be had therapy of this topic appropriate for an introductory graduate direction on combinatorics, and contains the real Robinson-Schensted-Knuth set of rules. additionally lined are connections among symmetric capabilities and illustration concept. An appendix by way of Sergey Fomin covers a few deeper elements of symmetric functionality concept, together with jeu de taquin and the Littlewood-Richardson rule. As in quantity 1, the workouts play an important position in constructing the cloth. There are over 250 workouts, all with suggestions or references to strategies, a lot of which crisis formerly unpublished effects. Graduate scholars and examine mathematicians who desire to observe combinatorics to their paintings will locate this an authoritative reference.

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**Additional info for Enumerative Combinatorics, Volume 2 (Cambridge Studies in Advanced Mathematics, Volume 62)**

**Example text**

However in some cases it is possible to solve the classiﬁcation problem. 1. 2-dimensional systems Let R be a PID and Σ = (A, B) a reachable m-input 2-dimensional linear system over R. Since U1 (B) = R there exist invertible matrices P and Q such that P BQ = 1 0 0 d ... 0 ... 0 where d is a generator of U2 (B). Considering the relevant actions of the feedback group we obtain the following result. P ROPOSITION 97. Let Σ = (A, B) be a 2-dimensional reachable linear system over a principal ideal domain R.

An−1 b)P −1 = c(A). Taking an adequate feedback matrix one has the following result. P ROPOSITION 82. Let Σ = (A, b) be a reachable single-input n-dimensional system. Then Σ is feedback equivalent to the system ⎛ ⎛ 0 1 ⎜ ⎜0 0 ⎜ ⎜. ⎜. Σ =⎜ ⎜A = ⎜ . ⎝ ⎝0 0 0 0 0 1 .. ··· ··· .. 0 0 ··· ··· ⎛ ⎞⎞ ⎞ 0 0 ⎜ 0 ⎟⎟ 0⎟ ⎜ ⎟⎟ .. ⎟ ˆ ⎜ .. ⎟ ⎟ , b = ⎜ . ⎟⎟ . ⎝ 0 ⎠⎠ 1⎠ 0 1 In particular, there exists a unique class of reachable single-input n-dimensional systems. The next objective is to study the multi-input case (m > 1).

Let R be a commutative ring and M an R-module. (i) M is projective if M is a direct summand of a free R-module or equivalently the functor Hom(M, − ) is exact. (ii) M is ﬂat if the functor M ⊗R − is exact. See [5, Chapters I and II] and [82] for the main properties of these modules. When M is ﬁnitely generated one has the following characterization. T HEOREM 19. Let M be a ﬁnitely generated R-module. Then: (i) M is ﬂat if and only if Mp is a free Rp -module for all prime ideals p of R. Linear algebra over commutative rings 13 (ii) M is projective if and only if M is ﬂat and the rank function rk : Spec(R) → Z p → rankRp Mp is continuous.