On logarithmic asymptotics for the number of restricted partitions in the exponential case

Congruences on the number of restricted m-ary partitions

A generalized Hardy-Ramanujan formula for the number of restricted integer partitions

In this paper, we expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let [Formula: see text] be the double sequences [Formula: see text] or [Formula: see text]. We associate double sequences [Formula: see text] and [Formula: see text], defined as the coefficients of [Formula: see text] [Formula: see text] These coefficients are related to the number of partitions [Formula: see text], plane partitions [Formula: see text] of [Formula: see text], and Fibonacci numbers [Formula: see text]. Let [Formula: see text] and let [Formula: see text]. Then the coefficients are log-concave at [Formula: see text] for almost all [Formula: see text] in the exponential (involving [Formula: see text]) and geometric cases (involving [Formula: see text]). The coefficients are not log-concave for almost all [Formula: see text] in both cases, if [Formula: see text]. Let [Formula: see text]. Then the log-concave property flips for almost all [Formula: see text].

We use the quantum statistical approach to estimate the number of restricted plane partitions of an integer n with the number of parts not exceeding some finite N. We use the analogy between this number theory problem and the enumeration of microstates of the ideal two-dimensional Bose gas. The numbers of restricted plane partitions calculated with the conjectured expression agree well with the exact values for n from 10 to 20.

There is an exact one-to-one correspondence between the number of (d - 1)-dimensional partitions of an integer and the number of directed compact lattice animals in d dimensions. Using enumeration techniques, we obtain upper and lower bounds for the number of multidimensional partitions (both restricted and unrestricted). We show that asymptotically the number of unrestricted (d - 1)-dimensional partitions of an integer n goes as exp. We also show that for restricted partitions in (d - 1) dimensions (with j dimensions finite, 0 < j < d-1), this number goes as , where is the extent of the lattice along the kth axis.

We study the correlators of irregular vertex operators in two-dimensional conformal field theory (CFT) in order to propose an exact analytic formula for calculating numbers of partitions, that is: 1) for given $N,k$, finding the total number $\lambda(N|k)$ of length $k$ partitions of $N$: $N=n_1+...+n_k;0<n_1\leq{n_2}...\leq{n_k}$. 2) finding the total number $\lambda(N)=\sum_{k=1}^N\lambda(N|k)$ of partitions of a natural number $N$ We propose an exact analytic expression for $\lambda(N|k)$ by relating two-point short-distance correlation functions of irregular vertex operators in $c=1$ conformal field theory ( the form of the operators is established in this paper): with the first correlator counting the partitions in the upper half-plane and the second one obtained from the first correlator by conformal transformations of the form $f(z)=h(z)e^{-{i\over{z}}}$ where $h(z)$ is regular and non-vanishing at $z=0$. The final formula for $\lambda(N|k)$ is given in terms of regularized ($\epsilon$-ordered) finite series in the generalized higher-derivative Schwarzians and incomplete Bell polynomials of the above conformal transformation at $z=i\epsilon$ ($\epsilon\rightarrow{0}$)

We study the correlators of irregular vertex operators in two-dimensional conformal field theory (CFT) in order to propose an exact analytic formula for calculating numbers of partitions, that is: 1) for given $N,k$, finding the total number $\lambda(N|k)$ of length $k$ partitions of $N$: $N=n_1+...+n_k;0<n_1\leq{n_2}...\leq{n_k}$. 2) finding the total number $\lambda(N)=\sum_{k=1}^N\lambda(N|k)$ of partitions of a natural number $N$ We propose an exact analytic expression for $\lambda(N|k)$ by relating two-point short-distance correlation functions of irregular vertex operators in $c=1$ conformal field theory ( the form of the operators is established in this paper): with the first correlator counting the partitions in the upper half-plane and the second one obtained from the first correlator by conformal transformations of the form $f(z)=h(z)e^{-{i\over{z}}}$ where $h(z)$ is regular and non-vanishing at $z=0$. The final formula for $\lambda(N|k)$ is given in terms of regularized ($\epsilon$-ordered) finite series in the generalized higher-derivative Schwarzians and incomplete Bell polynomials of the above conformal transformation at $z=i\epsilon$ ($\epsilon\rightarrow{0}$)

A method for generating restricted partitions of numbers is presented as an aid in helping the student construct three-arc cycles in certain directed graphs. Through this and similar devices the study of linear graph theory, as a whole, becomes more real to the student.

In this paper, we investigate the divisibility of the function b(n), counting the number of certain restricted 3-colored partitions of n. We obtain one Ramanujan type identity, which implies that b(3n + 2) ≡ 0 ( mod 3). Furthermore, we study the generating function for b(3n + 1) by modular forms. Finally, we find two cranks as combinatorial interpretations of the fact that b(3n + 2) is divisible by 3 for any n.

A note on the restricted partition function [formula omitted

A study of a state-dependent job admission policy in a computer system with restricted memory partitions

The theorems of Bachet–Bézout and Gauss are the cornerstones of elementary methods in number theory. Many applications of these results are studied, in particular Diophantine problems. The section Further Developments investigates the number of integer solutions of certain linear Diophantine equations, i.e. the number of certain restricted partitions of an integer.KeywordsLinear Diophantine EquationsDiophantine ProblemsInteger SolutionResidue Classes ModuloSaddle Point TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let P K, L (N) be the number of unordered partitions of a positive integer N into K or fewer positive integer parts, each part not exceeding L. A distribution of the form Ʃ/N≤x P K,L (N) is considered first. For any fixed K, this distribution approaches a piecewise polynomial function as L increases to infinity. As both K and L approach infinity, this distribution is asymptotically normal. These results are proved by studying the convergence of the characteristic function. The main result is the asymptotic behavior of P K,K (N) itself, for certain large K and N. This is obtained by studying a contour integral of the generating function taken along the unit circle. The bulk of the estimate comes from integrating along a small arc near the point 1. Diophantine approximation is used to show that the integral along the rest of the circle is much smaller.

Algorithm 262: Number of restricted partitions of N

A new test data reduction technique called accumulator compression testng (ACT) is proposed. ACT is an extension of syndrome testing. It is shown that the enumeration of errors missed by ACT for a unit under test is equivalent to the number of restricted partitions of a number. Asymptotic results are obtained for independent and dependent error modes. Comparison is made between signature analysis (SA) and ACT. Theoretical results indicate that with ACT a better control over fault coverage can be obtained than with SA. Experimental results are supportive of this indication. Built-in self test for processor environments may be feasible with ACT. However, for general VLSI circuits the complexity of ACT may be a problem as an adder is necessary.