FindStatFact 3: Spin and dinv adjustment on partition

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 \nabla(s_\lambda) [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 \lambda can be decomposed into border strips. For 0 \leq j < \lambda_1 let n_j be the length of the border strip starting at (\lambda_1-j,0).

  • The spin of \lambda 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 \sum (\lambda_1-1-j) where the sum ranges over all indices j such that n_j > 0.

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 (\lambda_1-j,0) crosses the vertical lines in the Ferrers shape exactly n_j-1-j 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.

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FindStatFact 2: Weak exceedances on Dyck paths

Who knows of a way to describe the number of weak exceedances of a permutation in terms of Dyck paths?

Go to http://www.findstat.org/St000213, click on search for statistic, and you will find out that

  • applying the inverse of the first fundamental transformation $\latex \phi$ given by sending the maximal entries in the cycles of a permutation \pi to the left-to-right maxima of \phi(\pi), then
  • sending $\phi(\pi)$ to its increasing tree T(\phi(\pi)), and finally
  • sending T(\phi(\pi)) to the Dyck path D by identifying an up step in D with a left subtree in T(\phi(\pi)), and a down step inĀ D with a right subtree in T(\phi(\pi)),

will send the number of weak exceedances of \pi to the number of peaks of a Dyck path.