#### Alladi Ramakrishnan Hall

#### Decorated staircases and applications

#### Arghya Mondal

##### IMSc

*Motivated from representation theory we have the following combinatorial problem. Consider the set of all decreasing sequences of real numbers of length n. We say two sequences $x=(x_1,..,x_n)$ and $y=(y_1,..,y_n)$ are equivalent if for each $1 \leq j < i \leq n$, both $x_i+x_j$ and $y_i+y_j$ are either positive or negative or zero. To each equivalence class we associate a combinatorial diagram called decorated staircase. We will show that the set of decorated staircases are in bijective correspondence with the set of equivalence classes and also count their number. The connection to certain representations of the*

Lie algebra $\mathfrak{so}^*(2n)$ will be stated.

Done