We find by applying MacMahons partition analysis that all magic labellings of the cube are of eight types, each generated by six basis elements. A combinatorial proof of this fact is given. The number of magic labellings of the cube is thus reobtaine
d as a polynomial in the magic sum of degree $5$. Then we enumerate magic distinct labellings, the number of which turns out to be a quasi-polynomial of period 720720. We also find the group of symmetry can be used to significantly simplify the computation.
A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $vin V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave remarkable
results on magic labellings, but the distinct labelling case is much more complicated. We consider the complete construction of all magic labellings of a given graph $G$. The idea is illustrated in detail by dealing with three regular graphs. We give combinatorial proofs. The structure result was used to enumerate the corresponding magic distinct labellings.
Let $L(m,n)$ denote Youngs lattice consisting of all partitions whose Young diagrams are contained in the $mtimes n$ rectangle. It is a well-known result that the poset $L(m,n)$ is rank symmetric, rank unimodal, and Sperner. A direct proof of this re
sult by finding an explicit order matching of $L(m,n)$ is an outstanding open problem. In this paper, we present an explicit order matching $varphi$ for $L(3,n)$ by several different approaches, and give chain tableau version of $varphi$ that is very helpful in finding patterns. It is surprise that the greedy algorithm and a recursive knead process also give the same order matching. Our methods extend for $L(4,n)$.
We give two proofs of the $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{vec{k}}(q,t)$ for $vec{k}=(k_1,k_2,k_3)$. One is by MacMahons partition analysis as we proposed; the other is by a direct bijection.
A polynomial $A(q)=sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0le a_1le cdots le a_kge a_{k+1} ge cdots ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= frac{1}{[n+m]} left[ m+n at
op nright]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $mle 5$ in a straightforward way.
Garsia and Xin gave a linear algorithm for inverting the sweep map for Fuss rational Dyck paths in $D_{m,n}$ where $m=knpm 1$. They introduced an intermediate family $mathcal{T}_n^k$ of certain standard Young tableau. Then inverting the sweep map is
done by a simple walking algorithm on a $Tin mathcal{T}_n^k$. We find their idea naturally extends for $mathbf{k}^pm$-Dyck paths, and also for $mathbf{k}$-Dyck paths (reducing to $k$-Dyck paths for the equal parameter case). The intermediate object becomes a similar type of tableau in $mathcal{T}_mathbf{k}$ of different column lengths. This approach is independent of the Thomas-Williams algorithm for inverting the general modular sweep map.
Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Ciglers Hankel determinant conjectures, which were proved
recently by Chang-Hu-Zhang using direct determinant computation. We find that shifted periodic continued fractions arise in our computation. We also discover and prove some new nice Hankel determinants relating to lattice paths with step set ${(1,1),(q,0), (ell-1,-1)}$ for integer parameters $m,q,ell$. Again shifted periodic continued fractions appear.
