This paper shows how the study of colored compositions of integers reveals some unexpected and original connection with the Invert operator. The Invert operator becomes an important tool to solve the problem of directly counting the number of colored compositions for any coloration. The interesting consequences arising from this relationship also give an immediate and simple criterion to determ...

starting from the 6th century a.h., and especially starting from the mongol era onward, along with the flourishing of historiography, historical shāhnāmeh-composition also became a point of interest, and shāhnāmehs appeared with the aim of recording historical events. in the composition of these shāhnāmehs, ferdowsi’s shāhnāmeh was like a model for the shāhnāmeh-writers, and due to this the spi...

In this paper, we give two new identities for compositions, or ordered partitions, of integers. These two identities are based on closely-related integer partition functions which have recently been studied. Thanks to the structure inherent in integer compositions, we are also able to extensively generalize both of these identities. Bijective proofs are given and generating functions are provid...

Composability is the capability to select and assemble simulation components in various combinations into simulation systems. The defining characteristic of composability is the ability to combine and recombine components. Composability exists in two forms, syntactic and semantic (also known as engineering and modeling). Syntactic composability is the implementation of components so that they c...

Concave compositions are compositions (i.e. ordered partitions) of a number in which the parts decrease up to the middle summand(s) and increase thereafter. Perhaps the most surprising result is for even length, concave compositions where the generating function turns out to be the quotient of two instances of the pentagonal number theorem with variations of sign. The false theta function disco...

Any Gray code for a set of combinatorial objects defines a total order relation on this set: x is less than y if and only if y occurs after x in the Gray code list. Let ≺ denote the order relation induced by the classical Gray code for the product set (the natural extension of the Binary Reflected Gray Code to k-ary tuples). The restriction of ≺ to the set of compositions and bounded compositio...

Recurrent neural networks have recently been used for learning to describe images using natural language. However, it has been observed that these models generalize poorly to scenes that were not observed during training, possibly depending too strongly on the statistics of the text in the training data. Here we propose to describe images using short structured representations, aiming to captur...

Introduction: The ureilites are an unusual class of carbon-rich, ultramafic, achondritic meteorites, which display both primitive and evolved characteristics. For example, their noble gases (which are carried largely in the diamond phase) display a planetary or fractionated signature, and their O isotopes retain a heterogeneity imparted from the solar nebula [1,2]. Paradoxically, though, the mi...

Recurrent neural networks can be trained to describe images with natural language, but it has been observed that they generalize poorly to new scenes at test time. Here we provide an experimental framework to quantify their generalization to unseen compositions. By describing images using short structured representations, we tease apart and evaluate separately two types of generalization: (1) g...