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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 $k$?

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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.

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