Set-Valued Young Tableaux and Product-Coproduct Prographs

Standard set-valued Young tableaux are a generalization of standard Young tableaux where cells can contain unordered sets of integers, with the added condition that every integer at position $(i,j)$ must be smaller that every integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore properties of standard set-valued Young tableaux with three rows and a fixed number of integers in every cell of each row (referred to as set-valued tableaux with row-constant density). Our primary focus is on standard set-valued Young tableaux with $1$ integer in each first-row cell, $k-1$ integers in each second-row cell, and $1$ integer in each third-row cell. For rectangular shapes $\lambda=n^3$, such tableaux are placed in bijection with closed $k$-ary product-coproduct prographs: directed plane graphs that correspond to finite compositions involving a $k$-ary product operator and a $k$-ary coproduct operator. That bijection is extended to three-row set-valued Young tableaux of non-rectangular and skew shape, and it is shown that a set-valued analogue of the Sch\"utzenberger involution on tableaux corresponds to $180$-degree rotation of the associated prographs. As a set-valued analogue of the hook-length formula is currently lacking, we also present direct enumerations of three-row standard set-valued Young tableaux for a variety of row-constant densities and a small number of columns. We then argue why the numbers of tableaux with the row-constant density $(1,k-1,1)$ should be interpreted as a one-parameter generalization of the three-dimensional Catalan numbers that mirrors the generalization of the (two-dimensional) Catalan numbers provided by the $k$-Catalan numbers.