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Recently, by the Riordans identity related to tree enumerations, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}(k+1)!(n+1)^{n-k} &=& (n+1)^{n+1}, end{eqnarray*} Sun and Xu derived another analogous one, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, end{eqnarray*} where $D_{k}$ is the number of permutations with no fixed points on ${1,2,dots, k}$. In the paper, we utilize the $lambda$-factorials of $n$, defined by Eriksen, Freij and W$ddot{a}$stlund, to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and another two algebraic proofs. Using the umbral representation of our generalized identity and the Abels binomial formula, we deduce several properties for $lambda$-factorials of $n$ and establish the curious relations between the generating functions of general and exponential types for any sequence of numbers or polynomials.
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To any rooted tree, we associate a sequence of numbers that we call the logarithmic factorials of the tree. This provides a generalization of Bhargavas factorials to a natural combinatorial setting suitable for studying questions around generalized factorials. We discuss several basic aspects of the framework in this paper. In particular, we relate the growth of the sequence of logarithmic factorials associated to a tree to the transience of the random walk and the existence of a harmonic measure on the tree, obtain an equidistribution theorem for factorial-determining-sequences of subsets of local fields, and provide a factorial-based characterization of the branching number of infinite trees. Our treatment is based on a local weighting process in the tree which gives an effective way of constructing the factorial sequence.
Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $frac{p(n-1)}{p(n)}left(1+frac{1}{n}right)>frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=sqrt[n]{p(n)/n}$ and $Delta$ be the difference operator respect to $n$. Desalvo and Pak pointed out that their approach to proving the log-concavity of $p(n)$ may be employed to prove a conjecture of Sun on the log-convexity of ${r(n)}_{ngeq 61}$, as long as one finds an appropriate estimate of $Delta^2 log r(n-1)$. In this paper, we obtain a lower bound for $Delta^2log r(n-1)$, leading to a proof of this conjecture. From the log-convexity of ${r(n)}_{ngeq61}$ and ${sqrt[n]{n}}_{ngeq4}$, we are led to a proof of another conjecture of Sun on the log-convexity of ${sqrt[n]{p(n)}}_{ngeq27}$. Furthermore, we show that $limlimits_{n rightarrow +infty}n^{frac{5}{2}}Delta^2logsqrt[n]{p(n)}=3pi/sqrt{24}$. Finally, by finding an upper bound of $Delta^2 logsqrt[n-1]{p(n-1)}$, we prove an inequality on the ratio $frac{sqrt[n-1]{p(n-1)}}{sqrt[n]{p(n)}}$ analogous to the above inequality on the ratio $frac{p(n-1)}{p(n)}$.
Assume $ k $ is a positive integer, $ lambda={k_1,k_2,...,k_q} $ is a partition of $ k $ and $ G $ is a graph. A $lambda$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ bigcup_{vin V(G)} L(v) $ can be partitioned into $ q $ subsets $ C_1cup C_2cupcdotscup C_q $ and for each vertex $ v $ of $ G $, $ |L(v)cap C_i|=k_i $. We say $ G $ is $lambda$-choosable if for each $lambda$-assignment $ L $ of $ G $, $ G $ is $ L $-colourable. In particular, if $ lambda={k} $, then $lambda$-choosable is the same as $ k $-choosable, if $ lambda={1, 1,...,1} $, then $lambda$-choosable is equivalent to $ k $-colourable. For the other partitions of $ k $ sandwiched between $ {k} $ and $ {1, 1,...,1} $ in terms of refinements, $lambda$-choosability reveals a complex hierarchy of colourability of graphs. Assume $lambda={k_1, ldots, k_q} $ is a partition of $ k $ and $lambda $ is a partition of $ kge k $. We write $ lambdale lambda $ if there is a partition $lambda={k_1, ldots, k_q}$ of $k$ with $k_i ge k_i$ for $i=1,2,ldots, q$ and $lambda$ is a refinement of $lambda$. It follows from the definition that if $ lambdale lambda $, then every $lambda$-choosable graph is $lambda$-choosable. It was proved in [X. Zhu, A refinement of choosability of graphs, J. Combin. Theory, Ser. B 141 (2020) 143 - 164] that the converse is also true. This paper strengthens this result and proves that for any $ lambda otle lambda $, for any integer $g$, there exists a graph of girth at least $g$ which is $lambda$-choosable but not $lambda$-choosable.
We study the central part of Lambda N and Lambda Lambda potential by considering the correlated and uncorrelated two-meson exchange besides the omega exchange contribution. The correlated two-meson is evaluated in a chiral unitary approach. We find that a short range repulsion is generated by the correlated two-meson potential which also produces an attraction in the intermediate distance region. The uncorrelated two-meson exchange produces a sizeable attraction in all cases which is counterbalanced by omega exchange contribution.
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