Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A partition of $n$ is a weakly decreasing sequence of natural numbers $\lambda = (\lambda_1, \lambda_2, \dots)$ such that $\sum \lambda_i = n$. Its length $l(\lambda)$ is the number of positive summands $\lambda_i$.

In exercise 3.19 of his book "The $q, t$-Catalan Numbers and the Space of Diagonal Harmonics" James Haglund defines a dinv-statistic on partitions as the number of cells $x$ in the Young diagram of $\lambda$ such that $leg(x) \leq arm(x) \leq leg(x) + 1$.

During my work on my master's thesis I discovered that $$ \sum_{\lambda} q^{dinv(\lambda)} = \sum_{\lambda} q^{l(\lambda)} $$ where the sum is taken over all partitions of $n$. My only way of proving this result uses the symmetry of the $q, t$-Catalan numbers, i.e., $C_n(q, t) = C_n(t, q)$.

I would like to know:

Is this result known to anyone else?

If so, is there a bijective proof, i. e., does anyone know a bijection mapping the partitions of $n$ with $dinv(\lambda) = k$ to partitions of $n$ with length $k$?

share|improve this question
add comment

1 Answer

up vote 6 down vote accepted

This formula appears in Exercise 1.103 of Enumerative Combinatorics, vol. 1, second ed. It was first proved by K. Liu, C. H. F. Yan, and J. Zhou, Sci. China, Ser. A 45 (2002), 420-431. A combinatorial proof was given by G. Warrington, J. Combinatorial Theory Ser. A 116 (2009), 379-403.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.