Fall 2014, UC Davis

Lectures: |
MWF 3:10-4:00pm in Bainer 1128
CRN 64084 |

Office hours: Wednesday 5:10-6pm | |

Instructor: |
Anne Schilling, MSB 3222, phone: 554-2326, anne@math.ucdavis.edu |

Text: |
Richard P. Stanley, "Enumerative Combinatorics, Volume I" Cambridge Studies in Advanced Mathematics 49, Cambridge University Press 1997. |

Prerequisites: |
MAT 145, 150 or equivalent; or permission by instructor |

Grading: |
Homework presentations:
Problems will be assigned regularly in class. Students are expected to work on these problems in groups of 2-3 students. Each group should present at least one or two homework solutions in class. |

Web: |
http://www.math.ucdavis.edu/~anne/FQ2014/245.html |

I. Introduction to counting (permutation statistics, twelvefold way)

II. Inclusion-Exclusion

III. Order (posets, lattices, Moebius inversion)

IV. Generating functions

The sequel to this course MAT 246 will cover symmetric functions and algebraic combinatorics.

- Motivation; example of a non-combinatorial and combinatorial proof (ch 1.1)
- Review of basic counting: sets and multisets (ch. 1.2)
- Cycles and inversions (ch. 1.3)
- Descents (ch. 1.4)
- Partitions and q-binomial coefficients (ch. 1.7)
- Partition identities (ch. 1.8)
- The twelvefold way (ch. 1.9)
- Rogers-Ramanujan identities
- Inclusion-exclusion (ch. 2.1, 2.2)
- Permutations with restricted position (ch. 2.3)
- Involutions (ch. 2.6)
- Posets (ch. 3.1)
- Lattices (ch. 3.3)
- Distributive lattices (ch. 3.4)
- Incidence algebras (ch. 3.6)
- Moebius inversion formula (ch. 3.7)
- Applications of Moebius inversion (ch. 3.8)
- Promotion and evacuation (ch. 3.10)
- Markov chain on linear extensions

An extension of Problem 17 can be found in:

Stanton, Dennis W.; White, Dennis E. A Schensted algorithm for rim hook tableaux. J. Combin. Theory Ser. A 40 (1985), no. 2, 211-247.

This is also related to n-cores and n-quotients of a partition.