A bijection for partitions with all ranks at least \(t\)
A nonincreasing sequence \(\lambda= (\lambda_1,\dots, \lambda_t)\) with all its `parts' \(\lambda_i\) positive integers and such that the weight \(|\lambda|= \lambda_1+\cdots+ \lambda_t\) of \(\lambda\) is \(n\), is called a partition of the natural number \(n\) and this fact is denoted by \(\lambda\vdash n\). For a given partition \(\lambda\) let \(\lambda^*= (\lambda^*_1,\dots, \lambda^*_s)\) be its conjugate partition (i.e., \(s= \lambda_1\) and \(\lambda^*_j\) is defined as the number of dots in the \(j\)th column of the Ferrers diagram for \(\lambda\)) and let \(d= d(\lambda)\) denote the length of a side of the Durfee square in the Ferrers diagram for \(\lambda\). The Durfee rectangle of a partition \(\lambda\) is defined to be the largest \(d^*\times (d^*+1)\)-rectangle contained in the Ferrers diagram of \(\lambda\); here \(d^*\) is called the height of the Durfee rectangle and denoted by \(d^*(\lambda)\). According to \textit{H. Gupta} [Fibonacci Q. 16, 548-552 (1978; Zbl 0399.10017)] the rank vector of \(\lambda\) is defined as the vector \(\overline r(\lambda)= (r_1,\dots, r_d)\) with \(r_i= \lambda_i- \lambda^*_i\) \((i= 1,\dots, d)\); the components \(r_i\) are called successive ranks of \(\lambda\). \({\mathfrak P}(n)\) denote the set of all partitions of \(n\), \({\mathfrak P}_b(n)\) the set of partitions with no part equal to \(b\), and \({\mathfrak R}(n)\) the set of those partitions in \({\mathfrak P}(n)\) that have all their successive ranks positive. It was observed by \textit{P. Erdős} and \textit{L. B. Richmond} [Combinatorica 13, 57-63 (1993; Zbl 0790.05008)] that the conjugate to any partition \(\lambda\) in \({\mathfrak R}(n)\) is graphical, i.e., \(\lambda\) is the degree sequence of some simple (undirected) graph. So \(|{\mathfrak R}(n)|\) serves for a lower bound on the number of graphical partitions \(\lambda\), \(\lambda\vdash n\). It has been known earlier that the results of \textit{G. E. Andrews} [Theory arithmetic functions, Lecture Notes Math. 251, 1-20 (1972; Zbl 0228.10015)] and the results of \textit{D. M. Bressoud} [J. Number Theory 12, 87-100 (1980; Zbl 0425.10014)] on the generalization of Rogers-Ramanujan identities imply that \(|{\mathfrak R}(n)|=|{\mathfrak P}_1(n)|\); direct proofs of this last identity were given by \textit{G. E. Andrews} [J. Stat. Plann. Inference 34, No. 1, 19-22 (1993; Zbl 0770.11044)] and by \textit{C. C. Rousseau} and \textit{F. Ali} [J. Comb. Theory, Ser. B 64, No. 2, 314-318 (1995; Zbl 0828.05006)]. In this paper the authors give a bijective proof for this identity. Namely, they find a bijection between the sets \({\mathfrak P}_1(n)\) and \({\mathfrak R}(n)\), which preserves the weight, the number of parts and the height of the Durfee rectangle of a given partition \(\lambda\) in \({\mathfrak P}_1(n)\) (Theorem 2). At the same time, it appears that the size of the corresponding Durfee square need not be preserved! Further, denote by \({\mathfrak R}_{\geq t}(n)\) the subset of \({\mathfrak P}(n)\) of those partitions \(\lambda\), the successive ranks of which are at least \(t\), and by \({\mathfrak R}_{=t}(n)\) the set of partitions of \(n\) whose minimum rank is exactly \(t\). Modifying their remarkable argument given in Section 2 which has sprung up from an idea by \textit{M. S. Cheema} and \textit{B. Gordon} [Duke Math. J. 31, 267-273 (1964; Zbl 0117.28204)] for two-rowed plane partitions, the authors prove in Section 3 that for \(t\leq 0\) there exists a bijection between the sets \({\mathfrak R}_{=t}(n)\) and \({\mathfrak R}_{\geq 1}(n- 1+t)\). This implies the existence of a bijection between \({\mathfrak R}_{\geq t}(n)\) and \({\mathfrak P}_{2-t}(n)\) (Theorem 4). The authors rephrase their results in terms of generating functions and prove (Theorem 6) that \[ {n\choose k}_q- q{n\choose k-1}_q= \sum_{\lambda\in{\mathfrak R}(n)\cap{\mathfrak L}(n, k)} q^{|\lambda|}+ 1 \] with the \(\left(\begin{smallmatrix} n\\ k\end{smallmatrix}\right)_q\) being the Gauss polynomials and \({\mathfrak L}(n,k)\) denoting the set partitions whose Ferrers diagrams lie in a \(k\times(n- k)\)-box. In this paper bijections are indicated also between the set \({\mathfrak R}(n)\cap{\mathfrak L}(n, k)\) and the subset \({\mathfrak A}(n,k)\) of \({\mathfrak L}(n,k)\) of those partitions which have their successive ranks all smaller than \(n- 2k\), or the subset \({\mathfrak Q}(n,k)\) of \({\mathfrak L}(n,k)\) of those partitions \(\lambda\) for which \(\lambda_1\geq k\), \(\lambda^*_d= d\) and \(\lambda_{i+1}\geq \lambda^*_i\) \((i= 1,\dots,d-1)\).
scientific article; zbMATH DE number 1212978
scientific article