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Peer-reviewedCC BY 4.0Mathematics

Certain Standard Young Tableaux of Non-Classical Shapes and Lattice Path Enumeration

  • Ping Sun1
  • Chen Xu1
  1. 1Department of Mathematics, Northeastern University, Shenyang 110819, China
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Abstract

A 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.

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Cite

BibTeXRIS
BibTeX
@article{sun2026certain,
  title = {{Certain Standard Young Tableaux of Non-Classical Shapes and Lattice Path Enumeration}},
  author = {Ping Sun and Chen Xu},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14203773},
  url = {https://www.mdpi.com/2227-7390/14/20/3773},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Certain Standard Young Tableaux of Non-Classical Shapes and Lattice Path Enumeration
AU  - Ping Sun
AU  - Chen Xu
PY  - 2026
DA  - 2026/10/17
JO  - Mathematics
DO  - 10.3390/math14203773
UR  - https://www.mdpi.com/2227-7390/14/20/3773
AB  - A 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.
SN  - 2227-7390
ER  -