# Undergrad-level combinatorics texts easier than Stanley's Enumerative Combinatorics?

I am an undergrad, math major, and I had basic combinatorics class before (undergrad level.) Currently reading Stanley's Enumerative Combinatorics with other math folks. We have found this book somewhat challenging~ Do you have any suggestions on other books to read? or books to help going thru Enumerative combinatorics?

• It would help to know (1) What you covered in your basic class, and (2) what you are finding difficult about Stanley. – Thomas Andrews Sep 27 '15 at 23:15
• We used Combinatorics and Graph Theory by Harris as the textbook, and now having a hard time of reading thru Stanley's book. Almost always got stuck with exercise problems... For instance, how many mxn matrices (each matrix only has 1 or 0 as the input) are there such that each row/column sums up to an even number/odd number... Basically I have no clue... – John Dynan Sep 27 '15 at 23:26
• But even the title of the book is probably too vague. Basically, you are expecting people to know both books, and then recommend a third book. The more information you give to potential answerers, the more likely you are to get good help. – Thomas Andrews Sep 27 '15 at 23:29
• The exercises in Stanley were not all intended to be solvable. There are several unsolved problems as exercises. Anything rated higher than 2 in difficulty is going to be extremely hard. He goes over the difficulty levels. 3 is a problem you could only expect a brilliant student to solve, 4 is at the level of a research paper, and 5 is unsolved. – Matt Samuel Aug 10 '16 at 2:23
• Here's a clue for that exercise about $0,1$-matrices. Try to show that, if an $m-1\times n-1$ submatrix is filled in arbitrarily with $0$s and $1$s, there's a unique way to fill in the remaining entries so that all the row and column sums are even. – bof Oct 10 '16 at 23:00

Here is a somewhat haphazard list of sources on algebraic combinatorics which appear to be suited to undergraduates (I have not personally read most of them, so I am making semi-educated guesses here). My notion of "algebraic combinatorics" includes such things as binomial coefficient identities, symmetric functions, lattice theory, enumerative problems, Young tableaux, determinant identities; it does not include graph theory (except for the parts of it that are secretly algebra) or extremal combinatorics.

General remarks:

• Combinatorics is a living subject, and so are the authors of many of the sources listed below. If you find errors, do let them know!

• If some of the links below are inaccessible, try adding https://web.archive.org/web/*/ before the link. For example, https://www.whitman.edu/mathematics/cgt_online/cgt.pdf would become https://web.archive.org/web/*/https://www.whitman.edu/mathematics/cgt_online/cgt.pdf. This will take you to an archived version of the link (assuming that archive.org has made such a version).

Textbooks/notes on algebraic combinatorics in general:

Stanley's EC (Enumerative Combinatorics) is supposed to be a challenging read for graduate students. In its (rather successful) attempt at being encyclopedic, it has very little space for details and leaves a lot to the reader. There are many other texts on combinatorics, and I suspect that the average among them will be easier to read than EC1 (although probably less "from the horse's mouth"). In no particular order:

I don't know these books/notes well enough to tell which of them are better suited for a first course (although I don't have any reasons to suspect any of them to be unsuitable), but it cannot hurt to try each of them and go as far as you can before meeting serious resistance. (And once you meet serious resistance, either keep going or try the next one.) Half of these are freely available (and so are the other half, if you search in the darker places).

Specific subjects:

Just a few so far...

Enumeration:

Young tableaux and representations of symmetric groups:

Monoids:

Combinatorics on words:

Chipfiring aka sandpiles:

Chip-firing is ostensibly about (a certain "game" on) graphs, but once you start studying it, algebraic structures quickly emerge. Thus, the subject is beloved by many combinatorialists that don't usually study graphs.

• The van Lint & Wilson is graduate level and not easy going. Bóna’s Introduction to Enumerative Combinatorics is very good; it would be a fine choice. The bits and pieces that I’ve seen of his other book are also good. The Brualdi is a bit drier but still readable. I just glanced through Guichard’s notes; they seem to be pretty evenly split between enumerative combinatorics and graph theory, and they’re a serious undergraduate introduction. – Brian M. Scott Sep 28 '15 at 16:14
• +1 for Loehr. The first edition was amazing. He recently published a second edition Combinatorics text with additional exposition on generating functions and algebraic combinatorics. – ml0105 Nov 10 '17 at 17:12
• @ml0105: Wow; where can I find this 2nd edition? (That said, the 1st edition I have already has generating functions, doing them better than any other source I know.) – darij grinberg Nov 10 '17 at 17:13
• I use Bona's "Walk Through Combinatorics" in my senior capstone, and I find it very well suited to that environment. His exposition is concise (borderline terse at times, but this isn't always a bad thing). He has numerous exercises, some with solutions and some without. The former make for excellent class discussion, while the latter make for excellent homework. The book gives the foundations of enumeration and graph theory and concludes with several chapters referred to as "horizons" (various special topics including Ramsey theory, probabilistic method, generating functions, and more). – Austin Mohr Nov 10 '17 at 17:26
• @darijgrinberg Check Amazon. The book is entitled “Combinatorics.” – ml0105 Nov 10 '17 at 18:28