Jeffrey C. Lagarias

This paper studies properties of the integer sequence $$\overline{\overline{G}}_n=\prod _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) _{\mathbb {Z,N}}$$ G ¯ ¯ n = ∏ k = 0 n n k Z , N which is analogous to $$\overline{G}_n=\prod _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) $$ G ¯ n = ∏ k = 0 n n k , the product of the elements of the n-th row of Pascal’s triangle. Here $$\left( {\begin{array}{c}n\\ k\end{array}}\right) _{\mathbb {Z,N}}$$ n k Z , N is an extended binomial coefficient, defined in the paper, constructed using an extended version of M. Bhargava’s theory of generalized factorials. In 1996 M. Bhargava introduced a generalization of the factorial function, $$n!_S=\prod _p\nu _n(S,p)$$ n ! S = ∏ p ν n ( S , p ) in terms of their prime factorization, and defines associated binomial coefficients. The last two authors extended Bhargava’s invariants further to define such invariants attached to each integer $$b\ge 2$$ b ≥ 2 . One obtains extended factorials and extended binomial coefficients, and the maximal extension defines extended factorials $$n!_{\mathbb {Z,N}}=\prod _{b\ge 2}b^{\alpha _n(\mathbb {Z},b)}$$ n ! Z , N = ∏ b ≥ 2 b α n ( Z , b ) including all $$b\ge 2$$ b ≥ 2 , with associated extended binomial coefficients $$\left( {\begin{array}{c}n\\ k\end{array}}\right) _{\mathbb {Z,N}}$$ n k Z , N , yielding $$\overline{\overline{G}}_n$$ G ¯ ¯ n . We have $$\overline{\overline{G}}_n=\prod _{b=2}^nb^{\overline{\nu }(n,b)}$$ G ¯ ¯ n = ∏ b = 2 n b ν ¯ ( n , b ) and the partial factorizations $$\overline{\overline{G}}(n,x)=\prod _{b=2}^{\lfloor x\rfloor }b^{\overline{\nu }(n,b)}$$ G ¯ ¯ ( n , x ) = ∏ b = 2 ⌊ x ⌋ b ν ¯ ( n , b ) . This paper shows $$\log \overline{\overline{G}}(n,\alpha n)$$ log G ¯ ¯ ( n , α n ) is well approximated by $$f_{\overline{\overline{G}}}(\alpha )n^2\log n+g_{\overline{\overline{G}}}(\alpha )n^2$$ f G ¯ ¯ ( α ) n 2 log n + g G ¯ ¯ ( α ) n 2 as $$n\rightarrow \infty $$ n → ∞ for limit functions $$f_{\overline{\overline{G}}}(\alpha )$$ f G ¯ ¯ ( α ) and $$g_{\overline{\overline{G}}}(\alpha )$$ g G ¯ ¯ ( α ) defined for all $$0\le \alpha \le 1$$ 0 ≤ α ≤ 1 . The remainder term has a power saving in n. The main results are deduced from study of functions $$\overline{A}(n,x)$$ A ¯ ( n , x ) and $$\overline{B}(n,x)$$ B ¯ ( n , x ) that encode statistics of the base b radix expansions of the integer n (and smaller integers), where the base b ranges over all integers $$2\le b\le x$$ 2 ≤ b ≤ x . Unconditional estimates of $$\overline{A}(n,x)$$ A ¯ ( n , x ) and $$\overline{B}(n,x)$$ B ¯ ( n , x ) are derived.

This paper extends Bhargava's theory of $\mathfrak{p}$-orderings of subsets $S$ of a Dedekind ring $R$ valid for prime ideals $\mathfrak{p}$ in $R$. Bhargava's theory defines for integers $k\ge1$ invariants of $S$, the generalized factorials $[k]!_S$, which are ideals of $R$. This paper defines $\mathfrak{b}$-orderings of subsets $S$ of a Dedekind domain $D$ for all nontrivial proper ideals $\mathfrak{b}$ of $D$. It defines generalized integers $[k]_{S,T}$, as ideals of $D$, which depend on $S$ and on a subset $T$ of the proper ideals $\mathscr{I}_D$ of $D$. It defines generalized factorials $[k]!_{S,T}$ and generalized binomial coefficients, as ideals of $D$. The extension to all ideals applies to Bhargava's enhanced notions of $r$-removed $\mathfrak{p}$-orderings, and $\mathfrak{p}$-orderings of order $h$.

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers $$p_i$$p i indexed by its parts i. We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size $$|\lambda |=n$$| λ | = n , their perimeter equaling n, and their largest part equaling n. We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to $$e^{\gamma } \log n$$e γ log n as $$n \rightarrow \infty $$n → ∞ .

An approximate divisor order is a partial order on the positive integers $\mathbb{N}^+$ that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on $\mathbb{N}^+$, produced using the floor function. A positive integer $d$ is a floor quotient of $n$, denoted $d \,\preccurlyeq_{1}\, n$, if there is a positive integer $k$ such that $d = \lfloor{n / k}\rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies a family of partial orders, the $a$-floor quotient relations $\,\preccurlyeq_{a}\,$, for $a \in \mathbb{N}^+$, which interpolate between the floor quotient order and the divisor order on $\mathbb{N}^+$. The paper studies the internal structure of these orders.

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset $S$ of $\mathbb{Z}$. Bhargava's factorials $k!_S$ are invariants, constructed using the notion of $p$-orderings of $S$ where $p$ is a prime. This paper defines $b$-orderings of any nonempty subset $S$ of $\mathbb{Z}$ for all integers $b\ge2$, as well as "extreme" cases $b=1$ and $b=0$. It defines generalized factorials $k !_{S,T}$ and generalized binomial coefficients $\binom{k+\ell}{k}_{S,T}$ as nonnegative integers, for all nonempty $S$ and allowing only $b$ in $T\subseteq\mathbb{N}$. It computes $b$-ordering invariants when $S$ is $\mathbb{Z}$ and when $S$ is the set of all primes.

A positive integer $d$ is a floor quotient of $n$ if there is a positive integer $k$ such that $d = \lfloor n/k \rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies the internal structure of this partial order and its M\"{o}bius function.

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers $p_i$ indexed by its parts $i$. We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size $|λ|=n$, their perimeter equaling $n$, and their largest part equaling $n$. We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to $e^γ \log n$ as $n \to \infty$.

The paper gives characterizations of the supernorm statistic $$\widehat{N} :\mathcal {P}\rightarrow \mathbb {N}^{+}$$N ^ : P → N + for partitions, where $$\mathcal {P}$$P is the set of integer partitions and $$\mathbb {N}^{+}$$N + is the positive integers. The supernorm statistic map is a bijection onto $$\mathbb {N}^{+}$$N + that defines a total order on partitions, the supernorm ordering, obtained by pulling back the additive total order on $$\mathbb {N}^{+}$$N + . The supernorm ordering refines two partial orders on partitions: the multiset inclusion order, whose image under $$\widehat{N}$$N ^ is the divisibility lattice on $$\mathbb {N}^{+}$$N + , and the Young’s lattice order. The paper shows that it is characterized by these two properties, additionally with the requirement of an order-isomorphism from the multiset inclusion order to the divisibility order. It is also characterized by these two properties, additionally with the requirement of mapping the partitions with exactly one part bijectively to the prime numbers. It presents a construction showing that the latter additional conditions are necessary for the characterization to hold.