We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.

Abstract: We introduce the immersion poset $$({\mathcal {P}}(n), \leqslant _I)$$ ( P ( n ) , ⩽ I ) on partitions, defined by $$\lambda \leqslant _I \mu $$ λ ⩽ I μ if and only if $$s_\mu (x_1, \ldots , x_N) - s_\lambda (x_1, \ldots , x_N)$$ s μ ( x 1 , … , x N ) - s λ ( x 1 , … , x N ) is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of $$GL_N({\mathbb {C}})$$ G L N ( C ) form an immersion pair, as defined by Prasad and Raghunathan [7]. We develop injections $$\textsf{SSYT}(\lambda , u ) \hookrightarrow \textsf{SSYT}(\mu , u )$$ SSYT ( λ , ν ) ↪ SSYT ( μ , ν ) on semistandard Young tableaux given constraints on the shape of $$\lambda $$ λ , and present results on immersion relations among hook and two column partitions. The standard immersion poset $$({\mathcal {P}}(n), \leqslant _{std})$$ ( P ( n ) , ⩽ std ) is a refinement of the immersion poset, defined by $$\lambda \leqslant _{std} \mu $$ λ ⩽ std μ if and only if $$\lambda \leqslant _D \mu $$ λ ⩽ D μ in dominance order and $$f^\lambda \leqslant f^\mu $$ f λ ⩽ f μ , where $$f^ u $$ f ν is the number of standard Young tableaux of shape $$ u $$ ν . We classify maximal elements of certain shapes in the standard immersion poset using the hook length formula. Finally, we prove Schur-positivity of power sum symmetric functions on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram [12].