共查询到20条相似文献,搜索用时 9 毫秒
1.
William J. Keith 《The Ramanujan Journal》2012,27(2):163-167
A theorem of Andrews equates partitions in which no part is repeated more than 2k−1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the generalizations that usually arise from such proofs. 相似文献
2.
William J. Keith 《The Ramanujan Journal》2014,35(1):157-164
It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4 and 5. A collection of conjectures includes two congruences modulo higher powers of 2 and a large family of “congruences with exceptions” for these and other regular partitions mod 3. 相似文献
3.
Rekha Natarajan 《Discrete Mathematics》2004,286(3):269-275
We present a bijection between non-crossing partitions of the set [2n+1] into n+1 blocks such that no block contains two consecutive integers, and the set of sequences such that 1?si?i, and if si=j, then si-r?j-r for 1?r?j-1. 相似文献
4.
We consider \(\text {pod}_3(n)\), the number of 3-regular partitions with odd parts distinct, whose generating function is where For each \(\alpha >0\), we obtain the generating function for where \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha }}\) if \(\alpha \) is even, \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha +1}}\) if \(\alpha \) is odd.
$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3(n)q^n=\frac{(-q;q^2)_\infty (q^6;q^6)_\infty }{(q^2;q^2)_\infty (-q^3;q^3)_\infty }=\frac{\psi (-q^3)}{\psi (-q)}, \end{aligned}$$
$$\begin{aligned} \psi (q)=\sum _{n\ge 0}q^{(n^2+n)/2}=\sum _{-\infty }^\infty q^{2n^2+n}. \end{aligned}$$
$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3\left( 3^{\alpha }n+\delta _\alpha \right) q^n, \end{aligned}$$
We show that the sequence {\(\text {pod}_3(n)\)} satisfies the internal congruences and
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$$\begin{aligned} \text {pod}_3(9n+2)\equiv \text {pod}_3(n)\pmod 9, \end{aligned}$$
(0.1)
$$\begin{aligned} \text {pod}_3(27n+20)\equiv \text {pod}_3(3n+2)\pmod {27} \end{aligned}$$
(0.2)
$$\begin{aligned} \text {pod}_3(243n+182)\equiv \text {pod}_3(27n+20)\pmod {81}. \end{aligned}$$
(0.3)
5.
Liuquan Wang 《The Ramanujan Journal》2018,47(1):99-115
Let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for \(b_{7}(n)\) modulo powers of 7: for \(n\ge 0\) and \(j\ge 1\), We also find some infinite families of congruences modulo 2 and 7 satisfied by \(b_{7}(n)\).
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$$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$
6.
Let t≥2 be an integer. We say that a partition is t-regular if none of its parts is divisible by t, and denote the number of t-regular partitions of n by b t (n). In this paper, we establish several infinite families of congruences modulo 2 for b 9(n). For example, we find that for all integers n≥0 and k≥0, $$b_9 \biggl(2^{6k+7}n+ \frac{2^{6k+6}-1}{3} \biggr)\equiv 0 \quad (\mathrm{mod}\ 2 ). $$ 相似文献
7.
Let \(b_{13}(n)\) denote the number of partitions of n such that no parts are divisible by 13. In this paper, we shall prove several infinite families of congruence relations modulo 2 for \(b_{13}(n)\). In addition, we will give an elementary proof of the parity result on broken 6-diamond partitions, which was established by Radu and Sellers. We also find a new congruence relation modulo 2 for the number of broken 6-diamond partitions of n. 相似文献
8.
9.
A. A. Jucys 《Lithuanian Mathematical Journal》1995,35(2):163-167
The one-to-one correspondence between the set of plane partitions withr rows andm columns and the set of matrices of nonnegative integers with the same numbers of rows and columns has been constructed.
Published in Lietuvos Matematikes Rinkinys, Vol. 35, No. 2, pp. 204–210, April–June, 1995. 相似文献
10.
《Discrete Mathematics》2020,343(6):111705
In this note, we give a simple extension map from partitions of subsets of to partitions of , which sends -distant -crossings to -distant -crossings (and similarly for nestings). This map provides a combinatorial proof of the fact that the numbers of enhanced, classical, and 2-distant -noncrossing partitions are each related to the next via the binomial transform. Our work resolves a recent conjecture of Zhicong Lin and generalizes earlier reduction identities for partitions. 相似文献
11.
Liuquan Wang 《The Ramanujan Journal》2017,44(2):343-358
Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that and
相似文献
$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$
$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$
12.
Jessica Striker 《Discrete Mathematics》2011,(21):331
We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices. 相似文献
13.
A τ-partition of a subset of a given p-set X can be extended in many different ways to a τ-partition of X. The Stirling number of the second kind S(p, τ) can be defined as the number of all τ-partitions of X, and can be regarded as the number of all possible extensions of the τ-partitions of all the τ-subsets of X. In this paper we study the number of extensions for a particular choice of τ-partitions of the m-subsets of X, with 1 ? m ? p. 相似文献
14.
The Ramanujan Journal - For a positive integer $$\ell $$ , let $$b_{\ell }(n)$$ denote the number of $$\ell $$ -regular partitions of a nonnegative integer n. Motivated by some recent conjectures... 相似文献
15.
Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that and
相似文献
$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$
$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$
16.
William Y.C. Chen Ira M. Gessel Catherine H. Yan Arthur L.B. Yang 《Journal of Combinatorial Theory, Series A》2008,115(6):1069-1076
We introduce a statistic pmaj(P) for partitions of [n], and show that it is equidistributed with cr2, the number of 2-crossings, over all partitions of [n] with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inv and maj. 相似文献
17.
We present a bijection for toroidal maps that are essentially 3-connected (3-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite toroidal maps with all faces of degree 4 except for a hexagonal root-face. We show that these maps are in bijection with certain well-characterized bipartite unicellular maps. Our bijection, closely related to the recent one by Bonichon and Lévêque for essentially 4-connected toroidal triangulations, can be seen as the toroidal counterpart of the one developed in the planar case by Fusy, Poulalhon and Schaeffer, and it extends the one recently proposed by Fusy and Lévêque for essentially simple toroidal triangulations. Moreover, we show that rooted essentially 3-connected toroidal maps can be decomposed into two pieces, a toroidal part that is treated by our bijection, and a planar part that is treated by the above-mentioned planar case bijection. This yields a combinatorial derivation for the bivariate generating function of rooted essentially 3-connected toroidal maps, counted by vertices and faces. 相似文献
18.
In this paper we give a bijection between the partitions of n with parts congruent to 1 or 4 (mod 5) and the partitions of n with parts differing by at least 2. This bijection is obtained by a cut-and-paste procedure which starts with a partition in one class and ends with a partition in the other class. The whole construction is a combination of a bijection discovered quite early by Schur and two bijections of our own. A basic principle concerning pairs of involutions provides the key for connecting all these bijections. It appears that our methods lead to an algorithm for constructing bijections for other identities of Rogers-Ramanujan type such as the Gordon identities. 相似文献
19.
20.
Periodica Mathematica Hungarica - In the paper, the authors present several new relations and applications for the combinatorial sequence that counts the possible partitions of a finite set with... 相似文献