Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function h [d] λ in (1.1) for any integer partition λ, and show that the transition matrix from h [d] λ to the power sum symmetric functions p λ is given bywhere D [d] and z are nonsingular diagonal matrices. Consequently, {h [d] λ } forms a basis of the ring Λ of symmetric functions. In addition, we show that the generating function H [d] (t) = n≥0 h [d] n (x)t n satisfieswhere ω is the involution of Λ sending each elementary symmetric function e λ to the complete homogeneous symmetric function h λ .

This paper extends Lewis's bijection (J. Combin. Theorey Ser. A 118, 2011) to a bijection between a more general class $\mathcal{L}(n,k,I)$ of permutations and the set of standard Young tableaux of shape $\langle (k+1)^n\rangle$, so the cardinality\[|\mathcal{L}(n,k,I)|=f^{\langle (k+1)^n\rangle},\]is independent of the choice of $I\subseteq [n]$. As a consequence, we obtain some new combinatorial realizations and identities on Catalan numbers. In the end, we raise a problem on finding a bijection between $\mathcal{L}(n,k,I)$ and $\mathcal{L}(n,k,I')$ for distinct $I$ and $I'$.

We will study the inversion statistic of $321$-avoiding permutations, and obtain that the number of $321$-avoiding permutations on $[n]$ with $m$ inversions is given by\[|\mathcal {S}_{n,m}(321)|=\sum_{b \vdash m}{n-\frac{\Delta(b)}{2}\choose l(b)}.\]where the sum runs over all compositions $b=(b_1,b_2,\ldots,b_k)$ of $m$, i.e.,\[m=b_1+b_2+\cdots+b_k \quad{\rm and}\quad  b_i\ge 1,\]$l(b)=k$ is the length of $b$, and $\Delta(b):=|b_1|+|b_2-b_1|+\cdots+|b_k-b_{k-1}|+|b_k|$. We obtain a new bijection from $321$-avoiding permutations to Dyck paths which establishes a relation on inversion number of $321$-avoiding permutations and valley height of Dyck paths.

In this paper, we study pattern avoidances of generalized permutations and show that the number of all generalized permutations avoiding π is independent of the choice of π ∈ S 3 , which extends the classic results on permutations avoiding π ∈ S 3 . Extending both Dyck path and Riordan path, we introduce the Catalan-Riordan path which turns out to be a combinatorial interpretation of the difference array of Catalan numbers. As applications, we interpret both Motzkin and Riordan numbers in two ways, via semistandard Young tableaux of two rows and generalized permutations avoiding π ∈ S 3 . Analogous to Lewis's method, we establish a bijection from generalized permutations to rectangular semistandard Young tableaux which will recover several known results in the literature.

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