# 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.

• 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 in $D$ 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.