11 problems

Let $\operatorname{Qsym}[t_1,t_2,\dots]$ denote the algebra of quasi-symmetric functions in countably many variables over the integers. Ditters' conjecture. The algebra … is a free…

Let $QSym(\mathbb Z)$ denote the universal ring of quasi-symmetric functions, identified with $J^{\vee}(\mathbb Z[y])=H^*(\Omega\Sigma\mathbb CP^{\infty};\mathbb Z)$, where $|y|=2$…

Let $P$ be a poset on $[n]=\{1,2,\ldots,n\}$, let $L(P)$ be its set of linear extensions, and define its free quasi-symmetric generating function by … Let the commutative image of…

Let $\alpha$ be a composition, and let $\lambda$ be the partition obtained by rearranging the parts of $\alpha$ into non-decreasing order. Write $F_\alpha$ for the fundamental quas…

Let $\sim$ be exotic Knuth equivalence, let $\mathbb{K}_\textsf{P}^{(\sim)}$ be the associated bialgebra, and let $\tilde\psi_{>|\leq}$ be the map to the completion of peak quasi-s…

Antipode morphism conjecture. There could be a morphism

Let $\A$ denote the completed algebra of symmetric functions, let $\hat{Z}$ be the completed zeta-type evaluation map, and let $\overline{I}_\beta$ be the closed ideal generated by…

Catalan socle conjecture. For every $n\geq 1$,

Let $\mathrm{QSym}$ be the algebra of quasi-symmetric functions, let $\mathrm{Sym}$ be the algebra of symmetric functions, and let $\bullet$ denote the product used to build identi…

Free dendriform trialgebra conjecture. The algebra ${\bf PQSym}^{(l)}$ is a free dendriform trialgebra.

Let $\operatorname{PolyMat}$ be the vector space of polymatroids, let $\operatorname{QSym}$ be the quasi-symmetric-function space, and let…