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 and at, 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.


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, 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.