Tip: Refresh the webpage if formulas are not shown correctly.

News

• 12/01/2016 Problem 32 is posted.

• 11/06/2016 Problem 31 is posted.

• 10/01/2016 Problem 30 is posted.

• 08/01/2016 Problem 29 is posted.

Enumeration, i.e., counting the number of elements of a finite set is an important topic in combinatorics. Project P, started in February 2015 by Ran Pan, is a collection of challenging (open) problems focused on enumeration. From January 2017, it is mainly maintained by Dun Qiu.

Many combinatorial objects are easy to understand but not that easy to count. Project P includes and does not limit to packings, partitions, paths, pattern matches, permutations, polygons, polynomials, polyominoes, posets, placements, probability and so on.

The answer to each problem is a sequence of integers. Our goal is to obtain the formula (closed-form or recurssive) of the sequence or the generating function of the sequence.

Since finding formulas or generating functions is very difficult sometimes (often), using computer programs to obtain some initial terms of the sequence also makes a fair contribution to solution. Therefore, students at any level from high schools to graduate schools and in any majors are welcome to participate in Project P.

At least one and up to three problems are posted in the first week of each month.

Problems labelled with letters from A to Z are warm-up exercise. Real problems begin from Problem 0.

Although all the warm-ups have been solved or could be found somewhere online, some of them are still worth thinking or working on. Warm-ups provide necessary background for real problems. People who are familiar with basic concepts in combinatorics may skip the warm-up section.

OEIS is short for the On-Line Encyclopedia of Integer Sequences which is an online database of integer sequences created by Neil Sloane in 1996.

Since the answer to each problem is an integer sequence, we could take advanatge of OEIS. Sometimes (perhaps often) it is easy to get some initial terms of the sequence while it is difficult to get the formula, then we look up the initial terms in OEIS. If there is a sequence that matches initial terms, we might be able to give a bijective proof between the combinatorial object in the problem and the one referred in the entry. Otherwise the answer is a new sequence to OEIS.

We strongly encourage people to submit their work. No matter initial terms, partial or full solution, any contribution is appreciated. Also people could submit their problems to Project P. Click here for more details of submission.

Special thanks to Prof. Remmel, Wilson Cheung, Lu Wang, Quang Tran Bach, Josh Tobin, Sinan Aksoy.