We’ll use this page to keep track of what has happened each day in class. It won’t contain any of the nitty-gritty details, but will instead serve to summarize what has transpired each day.

Week 1

  • Monday, January 14: First day! The first few minutes of class were devoted to me attempting to learn names. I think I got them all! Next, we did a very quick tour of BbLearn and the course website. The remainder of the class meeting was devoted to discussing isomorphisms as they occur in multiple mathematical contexts and we started to explore an example of a bijection between collections of "combinatorial objects" where some aspect of the structure is preserved.
  • Wednesday, January 16: After answering a few quick questions about the syllabus, I wrapped up the "prologue". Next, we had VM/DW and AM/JG2 present the recursive formula and closed form, respectively, for the minimum number of moves required to solve the $n$-disk Towers of Hanoi. Great first day of student presentations!
  • Friday, January 18: Busy day! We didn't discuss anything too ground breaking, but we covered a lot of territory. After a quick discussion of LaTeX and the first Weekly Homework assignment, we jumped into student presentations. We had JC, MG, MF, ND, and MS present Problems 3, 4, 5, 6, and 7, respectively. This left of us with plenty of time to get started on the next couple problems. After students worked in groups of 2-3, MR and VM presented two different approaches to Problem 8. I spent the remaining few minutes tying some ideas together.

Week 2

  • Monday, January 21: MLK Day, no class.
  • Wednesday, January 23: We had JG1, PG/AM/JW, JL, and JG3 present Problems 9, 10, 11, and 13, respectively. We had a nice discussion about the connection between Problem 8 (edges in complete graph) and Problem 9 (dominoes). Seeing multiple approaches to Problem 10 was cool.
  • Friday, January 25: We had FA, DW, RW, JG1, RW, and MS present Problems 14(2), 14(3), 14(4), 16, 18(via product principle), and 18(via induction), respectively.

Week 3

  • Monday, January 28: We kicked off with me summarizing Problems 19 and 20. Then we had JC, MF, and MG present Problems 22, 24, and 25, respectively. With the time we had left, we discussed permutations in their various guises.
  • Wednesday, January 30: We spent the first few minutes setting the stage and then jumped into student presentations. We had PG, JL, MR, AM and MG present Problems 27/29/30, 31(algebraic), 31(bijection), 32(algebraic), and 32(bijection), respectively.
  • Friday, February 1: JG3, RW, AM, and JG1 presented Problems 34, 36, 37, and 38, respectively. We very quickly summarized Problems 39 and 30 in the last couple minutes of class.

Week 4

  • Monday, February 4: I had my doubts about getting through everything today, but somehow we pulled it off. We had MS/AM, FA, JL, and PG present Problems 41, 42, 43/44, and 46, respectively. Along the way, I summarized Problem 45.
  • Wednesday, February 6: We had DW, ND, VM, and AM present Problems 47, 48/49/50, 51, and 52, respectively. We'll tackle Problem 53 on Friday.
  • Friday, February 8: After discussing the upcoming exam, we had MR present Problem 53. This was followed by a class discussion of all six parts of Problem 54. Next, we had LD and RW present parts 1/2 and 3 of Problem 55, respectively. We should have time to get caught up on Problem 56 on Monday.

Week 5

  • Monday, February 11: After discussing the upcoming exam, we cranked through all of the problems. We had JG1, FA, JG2/MS, and MF present Problems 56, 57, 58, and 60, respectively.
  • Wednesday, February 13: The students took the in-class portion of Exam 1.
  • Friday, February 15: I lectured today. We discussed Problems 61 and 62 and introduced generating functions.

Week 6

  • Monday, February 18: After handing back the in-class portion of Exam 1, we spent the entire class period discussing problems from both the in-class exam and the take-home exam.
  • Wednesday, February 20: Today we continued our discussion of generating functions. Along the way, we covered Problems 63-66.
  • Friday, February 22: Snow day!

Week 7

  • Monday, February 25: After discussing the plan for the upcoming week, we had FA and RW present both parts of Problem 67. Next, we discussed Problem 68.
  • Wednesday, February 27: Dr. Sieben covered for me while I was out of town. JC, JG3, MF, RW, ND/VM, and AM presented Problems 69(1,2), 69(3), 69(4), 69(4,5), 70, and 71, respectively.
  • Friday, March 1: After diving the class up into small groups, we had JL, MG, JG1 presented Problems 72, 73, and 74, respectively.

Week 8

  • Monday, March 4: We had JC, PG, MR, and AM present Problems 75(1), 75(2), 75(3), and 76, respectively.
  • Wednesday, March 6: We covered a lot of ground today! We had JG2, MS, JG3, AM, and MG present Problems 77, 78, 79, 80, and 81, respectively. With the few minutes we had left at the end of class, we briefly discussed Bell numbers.
  • Friday, March 8: After splitting the class up into several small groups, we had RW and MR present Problems 82 and 83, respectively. With the time we had left, we summarized Problem 84.

Week 9

  • Monday, March 11: We had JG1 and ND present Problems 85 and 86, respectively. This left us with plenty of time to get a jump on Problem 87.
  • Wednesday, March 13: We had RW/LD, VM, MS, and FA present Problems 87(3), 87(4), 93, and 94, respectively.
  • Friday, March 15: PG, ND, and AM presented Problems 97 and both parts of Problem 98, respectively.

Week 10

  • Monday, March 25: Today I lectured over Problems 99 and 100. We also got a start on Problem 101.
  • Wednesday, March 27: After a quick discussion of the upcoming exam, we had MF, MG, and MR present Problems 101, 102, and 103, respectively. I really like Problem 103!
  • Friday, March 29: I kicked off by quickly sketching a proof for Problem 107. The goal for the rest of the class meeting was to make progress on Problem 108, which I split up into five lemmas. I split the class up into five small groups, each of which was tasked with tackling one of the lemmas. We had MS, AM1, and MG present proofs for the first three lemmas.

Week 11

  • Monday, April 1: Picking up where we left off last week, we had PG and FA present the remaining two lemmas for Problem 108. Next, we had RW and MS present Problems 109 and 110, respectively.
  • Wednesday, April 3: The students took the in-class portion of Exam 2 (and there was chaos involving a typo on one of the problems).
  • Friday, April 5: We revisited Problems 107 and 109 and tied up some loose ends.

Week 12

  • Monday, April 8: After discussing a couple of the problems from the take-home exam, I continued lecturing. In particular, we wrapped up Problem 109 and started discussing inversions.
  • Wednesday, April 10: We revisited Problem 110 and then got most of the way through Problem 112.
  • Friday, April 12: We wrapped up Problem 112 and then returned to our discussion of inversions, which included Problem 116 and some other odds and ends.

Week 13

  • Monday, April 15: We had JL and MS present Problems 115 and 117, respectively. With the time we had left, I presented Problem 118.
  • Wednesday, April 17: We had JC, JG3, and AM present Problems 120, 121, and 122, respectively. We spent quite a bit of time on Problem 121, which generated lots of great discussion.
  • Friday, April 19: There was only one problem to discuss, but it took us the whole class session to get through it. We had FA, MS, RW, and AM present parts (2), (3), (4), and (5) of Problem 124, respectively.

Week 14

  • Monday, April 22: We kicked off with LD showing us a beautiful bijections from triangulations of an $(n+2)$-gon to planar binary trees with $n$ internal nodes. Next, I presented part (2) of Problem 125 and then FA presented a portion of part (3). With the time we had left, we kicked around some ideas about the remaining bits of part (3).
  • Wednesday, April 24: After (finally!) handing back the take-home exams, I summarized part (3) of Problem 125 and then MF presented part (2) of Problem 126. Next, MR proved that $L_{n,k}(q)$ is given by sum over permututations have descent set at most $k$. We will return to part (3) of Problem 126 on Friday.
  • Friday, April 26: We pretty much spent the whole class meeting discussing part (3) of Problem 126.

Week 15

  • Monday, April 29: I presented a fairly involved bijection for the first problem on Daily Homework 32.
  • Wednesday, May 1: We had MG and MR present the second problem from Daily Homework 32 and the problem for Daily Homework 33, respectively. In addition, I sketched an argument for the problem from Daily Homework 31.
  • Friday, May 3: Last day! I presented a cute bijection from $S_n$ to $S_n$ that mapped a permutation with $k$ inversions to a permutation with major index $k$.

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 411: Abstract Algebra
  MAT 690: Genome Combinatorics

About This Site

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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.