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 $k$?


1 Answer 1


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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