On Theta-palindromic Richness

In this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism $\Theta$. It generalizes the notion of a palindrome into a $\Theta$-palindrome -- a word invariant under $\Theta$. For languages closed under $\Theta$ we give the relation between $\Theta$-palindromic complexity and factor complexity. We generalize the notion of richness to $\Theta$-richness and we prove analogous characterizations of words that are $\Theta$-rich, especially in the case of set of factors invariant under $\Theta$. A criterion for $\Theta$-richness of $\Theta$-episturmian words is given together with other examples of $\Theta$-rich words.Comment: 14 page

Poly[aqua­(μ3-pyridazine-4-carboxyl­ato-κ2 O:O:O′)lithium]

The structure of the title compound, [Li(C5H3N2O2)(H2O)]n, is composed of centrosymmetric dimers in which two LiI ions are bridged by a carboxyl­ate O atom, each donated by a ligand, acting in a bidentate mode. The second carboxyl­ato O atoms bridge the dimers to LiI ions in adjacent dimers, forming mol­ecular layers parallel to (001). Each LiI ion is coordinated by two bridging carboxyl­ate O atoms, a bridging carboxyl­ate O atom donated by the adjacent dimer and an aqua O atom, resulting in a distorted tetra­hedral coordination geometry. The layers are held together by O—H⋯N hydrogen bonds in which coordinated water O atoms act as donors and ligand hetero-ring N atoms as acceptors

Generalized Thue-Morse words and palindromic richness

We prove that the generalized Thue-Morse word $\mathbf{t}_{b,m}$ defined for $b \geq 2$ and $m \geq 1$ as $\mathbf{t}_{b,m} = (s_b(n) \mod m)_{n=0}^{+\infty}$, where $s_b(n)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering simultaneously antimorphisms $\Theta \in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the terminology generalizing the notion of palindromic richness for more antimorphisms recently introduced by the author and E. Pelantov\'a, we show that $\mathbf{t}_{b,m}$ is $D_m$-rich. We also calculate the factor complexity of $\mathbf{t}_{b,m}$.Comment: 11 page

Poly[hydrazin-1-ium [diaqua­bis­(μ4-pyridazine-3,6-dicarboxyl­ato)trilithate] monohydrate]

The structure of the title compound, {(N2H5)[Li3(C6H2N2O4)2(H2O)2]·H2O}n, is composed of mol­ecular dimers, each built up of two symmetry-related LiI ions with distorted trigonal–bipyramidal coordinations bridged by two deprotonated ligand mol­ecules via their N,O-bonding sites. Doubly solvated LiI ions with a distorted tetra­hedral geometry link adjacent dimers, forming a polymer generated by bridging bidentate carboxyl­ato O atoms to LiI ions in adjacent dimers, forming anionic layers parallel to the ac plane with monoprotonated hydrazinium cations and crystal water mol­ecules positioned between them. The layers are held together by an extended system of hydrogen bonds in which the hydrazinium cations and coordinated and crystal water mol­ecules act as donors and carboxyl­ate O atoms act as acceptors

Poly[di-μ2-aqua-μ2-(5-methyl­pyrazine-2-carboxyl­ato)-(5-methyl­pyrazine-2-carboxyl­ato)-μ3-nitrato-trilithium]

The asymmetric unit of the title compound, [Li3(C6H5N2O2)2(NO3)(H2O)2]n contains three LiI ions, two ligand anions, two water mol­ecules and a nitrate anion. Related by a centre of inversion, they form a centrosymmetric mol­ecular cluster in which one of the LiI ions shows trigonal–bipyramidal and the other two distorted tetra­hedral coordination. LiI ions are bridged by water O atoms and carboxyl­ate O atoms donated by one of the ligands. The clusters, bridged by two nitrato O atoms, form mol­ecular columns along [010], which are held together by O—H⋯O and O—H⋯N hydrogen bonds and π–π inter­actions [centroid–centroid distances = 3.694 (1) and 3.796 (1) Å]