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We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $$w_{n,k,m}$$ is the number of Dyck paths of semilength $$n$$ with $$k$$ occurrences of $UD$ and $$m$$ occurrences of $UUD$, then $$w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $$w_{2k+1,k,m}$$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $$w_{n,k,m}$$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $$w_{n,k_{1},k_{2},\ldots, k_{r}}$$ counting the semilength-$$n$$ Dyck paths with $$k_{1}$$ $UD$-factors, $$k_{2}$$ $UUD$-factors, $$\ldots$$, and $$k_{r}$$ $$U^{r}D$$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $$W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$$, including real-rootedness, $$\gamma$$-positivity, and a symmetric decomposition.
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Linear maps preserving the Lorentz spectrum: the $2 \times 2$ case
In this paper, a complete description of the linear maps $$\phi:W_{n}\rightarrow W_{n}$$ that preserve the Lorentz spectrum is given when $n=2$, and $$W_{n}$$ is the space $$M_{n}$$ of $$n\times n$$ real matrices or the subspace $$S_{n}$$ of $$M_{n}$$ formed by the symmetric matrices. In both cases, it has been shown that $$\phi(A)=PAP^{-1}$$ for all $$A\in W_{2}$$, where $$P$$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lorentz cone). These results extend to $n=2$ those for $$n\geq 3$$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $$n\geq3,$$ due to the fact that the Lorentz cone in $$\mathbb{R}^{2}$$ is polyedral, contrary to what happens when it is contained in $$\mathbb{R}^{n}$$ with $$n\geq3.$$ Thus, the study of the Lorentz spectrum preservers on $$W_n = M_n$$ also follows from the known description of the Pareto spectrum preservers on $$M_n$$.
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- Award ID(s):
- 1850663
- PAR ID:
- 10391175
- Date Published:
- Journal Name:
- The Electronic Journal of Linear Algebra
- ISSN:
- 1081-3810
- Page Range / eLocation ID:
- 317 to 330
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.13001/ela.2022.6925
