An online seminar for presentation of open problems with combinatorial flavor. This includes problems within the field of Mathematics, Theoretical Computer Science, Statistics, etc. Each of the talks at the CROP seminar aims to lead to new collaborations and obtaining of new results.

Format: The total duration of each talk (including questions) is not more than 30 minutes. Most of the talks are 10 to 20 minutes with a few minutes for questions at the end. Each attendee can reach out to the presenter after the talk and the presenter can form a working group with all interested people. A useful resource here might be this file on supercollaboration.

Join our mailing list: Subscribe here. You will receive only one email two days prior to each talk.

Time of the seminar [FALL 2022]: Friday, 12:15pm (Eastern Time, i.e., New York time) ; Usually, we meet bi-weekly.

Zoom Link: https://rutgers.zoom.us/j/98270180477?pwd=VkpsRkg2ZWo4cDU5MTk0bFpzaENSZz09

Passcode: 112515

*Abstract*: Most of the permutation patterns literature considers avoidance of various patterns in a set of permutations of given size. However, in many real-life situations, we do not observe the whole permutation at the beginning, but we see its elements one by one. Respectively, we often stop looking at the new elements of the permutation once certain condition holds for the observed prefix (e.g., think of the popular *secretary problem*). How many of these prefixes do not have an increasing subsequence of length 3? In general, how many prefixes avoid a given permutation pattern? Some computer simulations give promising coincidences with several OEIS sequences for various stopping rules.

Zoom Link: https://rutgers.zoom.us/j/98270180477?pwd=VkpsRkg2ZWo4cDU5MTk0bFpzaENSZz09

Passcode: 112515

Zoom Link: https://rutgers.zoom.us/j/98270180477?pwd=VkpsRkg2ZWo4cDU5MTk0bFpzaENSZz09

Passcode: 112515

Zoom Link: https://rutgers.zoom.us/j/98270180477?pwd=VkpsRkg2ZWo4cDU5MTk0bFpzaENSZz09

Passcode: 112515

Zoom Link: https://rutgers.zoom.us/j/98270180477?pwd=VkpsRkg2ZWo4cDU5MTk0bFpzaENSZz09

Passcode: 112515

In this talk, we will explore the analogous problem (to that of Alon), where edges of the complete k-uniform hypergraph may appear in at least one and at most l of the complete k-partite k-uniform hypergraphs. Although the problem statement is relatively straightforward, there has been very little progress on this problem. In particular, the case where k=3 and l=2 is still wide open and no non-trivial bounds have been shown.

Zoom Link: https://rutgers.zoom.us/j/98270180477?pwd=VkpsRkg2ZWo4cDU5MTk0bFpzaENSZz09

Passcode: 112515

More concretely, for a finite word w, the anagraph AG(w) is a graph with vertex set all the anagrams of w. Two vertices are adjacent if and only if the corresponding words have different letters at every position. In our work, we determine sufficient and necessary conditions for the connectivity of anagraphs and conjecture that the same conditions also imply Hamiltonicity. This conjecture is based on the Cayley-graph perspective of anagraphs and casts the problem into the broader framework of understanding when Cayley graphs are Hamiltonian.