Standard Set-Valued Young Tableaux and Product-Coproduct Prographs

Standard set-valued Young tableaux are a generalization of standard Young tableaux where cells can contain more than one integer. Unlike standard Young tableaux, there is no known method to count the number of distinct set-valued tableaux of arbitrary shape. We construct bijections between standard set-valued tableaux and k-ary product-coproduct prographs, a generalization of k-ary trees, where internal vertices may be interpreted as either a k-ary product or k-ary coproduct. We present a bijection between 3-row rectangular tableaux that have k−1 integers in each middle row cell, and k-ary prographs with n products and coproducts, and then generalize our bijection to non-rectangular tableaux. We use this bijection to count the number of tableaux for small n. Furthermore, we investigate various intuitive operations on prographs and the corresponding tableaux. Finally, we define an analogue of the Schutzenberger involution for standard set-valued tableaux and show it corresponds to a 180-degree rotation on prographs