DOI: 10.3390/math14203773 ISSN: 2227-7390
Certain Standard Young Tableaux of Non-Classical Shapes and Lattice Path Enumeration
Ping Sun, Chen XuA cell (i,j) of a Young diagram is called an inner cell if all four of its neighbours (i±1,j) and (i,j±1) also belong to the diagram. A standard Young tableau (SYT) of a hollow shape is one whose Young diagram contains no inner cells. In this paper, we establish a correspondence between hollow SYTs and {(1,0),(0,1)}-lattice paths with boundary restrictions. Using this correspondence, we derive enumeration formulas for two families of hollow SYTs: hollow rectangles truncated by the shifted staircase shape δk, and hollow shifted strips of constant width. Conversely, we obtain new enumerative identities for lattice paths with rectangular, trapezoidal, and periodic diagonal stair-step boundaries via the bijection with hollow SYTs.