Journal of Combinatorics

Volume 13 (2022)

Number 3

Eulerian central limit theorems and Carlitz identities in positive elements of classical Weyl groups

Pages: 333 – 356



Hiranya Kishore Dey (Department of Mathematics, IIT Bombay, Powai, Mumbai, India)

Sivaramakrishnan Sivasubramanian (Department of Mathematics, IIT Bombay, Powai, Mumbai, India)


The Eulerian numbers counting descents in the symmetric group $\mathcal{S}_n$ are known to have a Central Limit Theorem. Recently, Fulman, Petersen, Kim and Lee showed similar results over the alternating group $\mathcal{A}_n$. We show similar Central Limit Theorem like results when one samples permutations uniformly from $\mathcal{A}_n$ and considers excedances. We extend our results to the case when we sample over the positive part of Classical Weyl Groups. For the type D Coxeter groups, Borowiec and Mlitkowski recently defined a new type D Eulerian number and we show Central Limit theorems when one samples this new Eulerian statistic over the positive part of $\mathcal{D}_n$. Our results give natural refinements of Carlitz type identities as well.


central limit theorems, Eulerian numbers, Carlitz identity, classical Weyl groups

2010 Mathematics Subject Classification

Primary 05A16, 05E99. Secondary 60F05.

The first author acknowledges support from a CSIR-SPM Fellowship. The second author thanks Professor Alladi Subramanyam for illuminating discussions on CLTs. Support from the project grant SERB/F/252/2019-2020 given by the Science and Engineering Research Board (SERB), India is also acknowledged.

Received 30 November 2020

Accepted 22 March 2021

Published 31 March 2022