Timeline for Bijection directly from (n,n+1)-core partitions to parking functions?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 31, 2017 at 17:08 | comment | added | coolpapa | This answer contains most of what is needed - thanks very much for your help! The last step is described in my answer below. | |
| May 30, 2017 at 17:12 | comment | added | coolpapa | I admit this idea is half-baked, but it might work. Parking functions are an indexing of maximal chains of non-crossing partitions (Stanley here: combinatorics.org/ojs/index.php/eljc/article/view/v4i2r20/pdf), and the Athanasiadis-Linusson bijection uses non-nesting partitions of a very particular type. I wonder if the non-crossing/non-nesting correspondence can be brought to bear somehow to make a circuitous link between Pak-Stanley and Athanasiadis-Linusson? | |
| May 30, 2017 at 16:47 | comment | added | Christian Stump | Oh, I see. I was missing that indeed. | |
| May 30, 2017 at 16:45 | comment | added | coolpapa | However, This is where the Pak-Stanley and Athanasiadis-Linusson bijections differ. P-S assigns the parking function (1,2,3) to the dominant alcove of the Shi arrangement farthest from the origin, and (1,1,1) to the fundamental alcove. A-L does the exact opposite. So I am looking for a direct way to turn the 3-core $(3,1,1)$ into the parking function $(1,1,1)$, and the 3-core $\emptyset$ into the parking function $(1,2,3)$. | |
| May 30, 2017 at 16:40 | comment | added | coolpapa | Thank you very much for the pointers. But I think Anderson's bijection matches up with the Pak-Stanley labelling of the Shi arrangement. I'll summarize my thinking briefly here in case I've done something incorrectly. For $(3,4)$-core partitions, Anderson tells us to look at the integer lattice: $\begin{array}{cccc} 1 & -2 & -5 & -8 \\ 5 & 2 & -1 & -4 \end{array}$. Then for, say, the partition (3,1,1), she would mark the hook numbers 5,2,1. That gives the Dyck path that alternates up and down steps, which corresponds to the parking function (1,2,3), since it must have no repeated values. | |
| May 30, 2017 at 6:25 | history | answered | Christian Stump | CC BY-SA 3.0 |