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We study the $\{1234, 3412\}$ pattern-replacement equivalence relation on the set $S_n$ of permutations of length $n$, which is conceptually similar to the Knuth relation. In particular, we enumerate and characterize the nontrivial equivalence classes, or equivalence classes with size greater than 1, in $S_n$ for $n \geq 7$ under the $\{1234, 3412\}$-equivalence. This proves a conjecture by Ma, who found three equivalence relations of interest in studying the number of nontrivial equivalence classes of $S_n$ under pattern-replacement equivalence relations with patterns of length $4$, enumerated the nontrivial classes under two of these relations, and left the aforementioned conjecture regarding enumeration under the third as an open problem., Comment: 13 pages, 0 figures

We extend past results on a family of formal power series $K_{n, \Lambda}$, parameterized by $n$ and $\Lambda \subseteq [n]$, that largely resemble quasisymmetric functions. This family of functions was conjectured to have the property that the product $K_{n, \Lambda}K_{m, \Omega}$ of any two functions $K_{n, \Lambda}$ and $K_{m, \Omega}$ from the family can be expressed as a linear combination of other functions from the family. In this paper, we show that this is indeed the case and that the span of the $K_{n, \Lambda}$'s forms an algebra. We also provide techniques for examining similar families of functions and a formula for the product $K_{n, \Lambda}K_{m, \Omega}$ when $n=1$., Comment: 16 pages, 0 figures

Neutron scattering techniques are used to investigate the crystalline and magnetic structure of LuFe$_{0.75}$Mn$_{0.25}$O$_{3}$ in bulk polycrystalline form. We find that the crystalline structure is described by the hexagonal P6$_{3}$cm space group similar to that of thin-film LuFeO$_{3}$, and that the system orders antiferromagnetically below T$_{N}$ = 134 K. Inelastic neutron scattering reveals nearest-neighbor superexchange parameters that are enhanced relative to LuMnO$_{3}$. The observation of significant diffuse scattering above T$_{N}$ demonstrates the frustrated nature of the system; comparisons with similar materials suggest the ground state magnetic configuration is sensitive to local crystallographic distortions., Comment: 15 pages, 5 figures