This FindStatFact is a simple observation we made a few weeks ago, which we think shows one strength of the FindStat project very well:
The paper Loehr, N. A., Warrington, G. S. Nested quantum Dyck paths and [MathSciNet, arXiv] defines the two statistics spin and dinv adjustment on integer partitions. These are also discussed in an appendix of the book Haglund, J. The q,t-Catalan numbers and the space of diagonal harmonics [MathSciNet]. We refer to these two references for the definitions. These can as well be found at www.FindStat.org/St000319 and at www.FindStat.org/St000320, respectively.
The Ferrers shape of an integer partition can be decomposed into border strips. For let be the length of the border strip starting at .
- The spin of is defined to be the total number of crossings of these border strips with the vertical lines in the Ferrers shape.
- The dinv adjustment is defined as where the sum ranges over all indices such that .
When we added them to FindStat recently, the search engine told us that these appear to coincide. And indeed, it is obvious that the border strip starting at crosses the vertical lines in the Ferrers shape exactly times.
Multiple people have looked at these statistics defined on the same page in a well-received paper. But no one had a reason to check whether or not these statistics coincide, so this obvious coincidence stayed undiscovered since 2007.