Warnaar's bijection and colored partition identities, I

We provide a general and unified combinatorial framework for a number of colored partition identities, which include the five, recently proved analytically by B. Berndt, that correspond to the exceptional modular equations of prime degree due to H. Schroeter, R. Russell and S. Ramanujan. Our approach generalizes that of S. Kim, who has given a bijective proof for two of these five identities, namely the ones modulo 7 (also known as the Farkas-Kra identity) and modulo 3. As a consequence of our method, we determine bijective proofs also for the two highly nontrivial identities modulo 5 and 11, thus leaving open combinatorially only the one modulo 23.Comment: Contains the first portion of the first author's MIT senior thesis (2011). Some minor revisions with respect to the previous version. To appear in JCT

Ir_urfs_vf: Image Recommendation with User Relevance Feedback Session and Visual Features in Vertical Image Search

In recent years, online shopping has grown exponentially and huge number of images are available online. Hence, it is necessary to recommend various product images to aid the user in effortless and efficient access to the desired products. In this paper, we present image recommendation framework with user relevance feedback session and visual features (IR_URFS_VF) to extract relevant images based on user inputs. User feedback is retrieved from image search history with clicked and un-clicked images. Image features are computed off-line and later used to find relevance between images. The relevance between images is determined by cosine similarity and are ranked based on clicked frequency and similarity score between images. Experiments results show that IR_URFS_VF outperforms CBIR method by providing more relevant ranked images to the user input query

Color Partition Identities Arising from Ramanujan's Theta-Functions

We establish several partition identities with distinct colors that arise from Ramanujan’s theta-function identities and formulas for multipliers in the theory of modular equations. Also, we deduce few partition congruences as a corollary of some partition identities

A partition identity and the universal mock theta function g2g_2

We prove analytic and combinatorial identities reminiscent of Schur's classical partition theorem. Specifically, we show that certain families of overpartitions whose parts satisfy gap conditions are equinumerous with partitions whose parts satisfy congruence conditions. Furthermore, if small parts are excluded, the resulting overpartitions are generated by the product of a modular form and Gordon and McIntosh's universal mock theta function. Finally, we give an interpretation for the universal mock theta function at real arguments in terms of certain conditional probabilities.Comment: 10 page

Warnaar’s bijection and colored partition identities, II

In our previous paper (J. Comb. Theory Ser. A 120(1):28–38, 2013), we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the Schröter, Russell, and Ramanujan type. The goal of this paper is to use the master bijection of Sandon and Zanello (J. Comb. Theory Ser. A 120(1):28–38, 2013) to show combinatorially several new and highly nontrivial colored partition identities. We conclude by listing a number of further interesting identities of the same type as conjectures.Massachusetts Institute of Technology. Dept. of Mathematics