An n-color composition is a colored composition in which a part of size m may come in m colors. This paper gives a new set of n-color-type compositions that admits exhaustive conjugation of its members. Previous attempts at conjugation of n-color compositions have yielded partial results at best. Instead of importing the coloring scheme previously used for partitions, we apply colors directly to the parts of compositions while treating any maximal string of ones as a single part under color assignment. This leads to the definition of n-color compositions of the second kind. As with ordinary compositions, a conjugate may be found using equivalent techniques: symbolic algebra, zig-zag graphs, and line graphs. We conclude with a derivation of the relevant enumeration formulas.

We consider properties of overpartitions that are simultaneously `-regular and & micro;-regular, where ` and & micro; are positive relatively prime integers. We prove a seven-way combinatorial identity related to these overpartitions. We also prove several congruence properties satisfied by this class of partitions (and a further related class) using both generating functions and modular forms with Radu's algorithm.

George Andrews [\emph{Bull. Amer. Math. Soc.}, 2007, 561--573] introduced the idea of a \emph{signed partiton} of an integer; similar to an ordinary integer partitions, but where some of the parts could be negative. Further, Andrews reinterpreted the classical G\"ollnitz--Gordon partition identities in terms of signed partitions. In the present work, we provide interpretations of the sum sides of Rogers--Ramanujan type identities, including a new signed partition interpretation of the G\"ollnitz--Gordon identities, different from that of Andrews. Both analytic and bijective proofs are presented.

Many classical q-series identities, such as the Rogers–Ramanujan identities, yield combinatorial interpretations in terms of integer partitions. Here we consider algebraically manipulating some of the classical q-series to yield natural combinatorial interpretations in terms of overpartitions. Bijective proofs are supplied as well.

In his classic text, Combinatory Analysis, MacMahon defined a perfect partition of a positive integer n as a partition whose parts contain exactly one partition of every positive integer not exceeding n. In this paper we apply the same definition to overpartitions which are integer partitions with the additional property that the final occurrence of each part may be overlined. It turns out that perfect overpartitions are enumerated by ordered factorization functions in which the occurrence of 2 as a factor determines the presence of an overlined part.

We introduce $n$-multicolor compositions which generalize the $n$-color compositions that Agarwal first defined twenty-two years ago. A summand may bear a set of many integer-valued colors which it bounds from above. We enumerate such multicolored compositions using generating functions. Then we isolate the conjugable class of multicolored compositions known as cracked compositions. A concise exposition of the conjugation of $n$-color compositions is presented in the classical tradition of Percy Alexander MacMahon (1854 - 1929). Our set of conjugable $n$-color compositions turn out to be considerably larger than previously known ones.

In this paper, we enumerate classes of partitions of [n] = {1, ... , n} in which the singleton blocks are colored using a variable or fixed number of colors. We consider, more generally, the distribution of the statistic recording the number of colored singletons on r-partitions of [r + n] in which only singletons from [r + 1, r + n] may be colored. Among our results, it is shown by algebraic and bijective arguments that the number of partitions of [n] in which a singleton block {x} can come in one of x colors for each x is given by the n-th row sum of Lah numbers, yielding a new combinatorial interpretation for this sequence. Also, we show that the partitions of [n] in which each singleton is assigned one of s + 1 colors where s is fixed are equinumerous with the set of s-partitions of [s +n]. Generalizations in terms of r-partitions of both of these results and others are demonstrated.

We study the conjugation of overpartitions and give the generating function for the number of self-conjugate overpartitions of an integer. Following the recent introduction of over q-binomial coefficients, we obtain the over q-analogue of the Chu-Vandermonde identity. Consequently a new generating function for the number of overpartitions is proved. We also give a new over q-analogue of the Chu-Vandermonde identity.

Sylvester’s theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of the theorem characterizes the partitions of a number as a sum of arithmetic progressions with positive terms. In addition to synthesizing and rediscovering known results, the method we propose allows us to state a second generalization and characterize the partitions of a number into parts whose differences between consecutive parts form an arithmetic progression. To achieve this, we will analyze the set of divisors in arithmetics that modify the usual definition of the multiplication operation between two integers. As we will see, symmetries arise in the set of divisors based on two parameters: t1, being even or odd, and t2, congruent to 0, 1, or 2 (mod 3). This approach also leads to a unique representation result of the same nature as Sylvester’s theorem, i.e., a power of 3 cannot be represented as a sum of three or more terms of a positive integer sequence such that the differences between consecutive terms are consecutive integers.

In analogy with the semi-Fibonacci partitions studied recently by Andrews, we define semi-\( m \)-Pell compositions. We find that these are in bijection with certain weakly unimodal \( m \)-ary compositions. We give generating functions, bijective proofs, and a number of unexpected congruences for these objects. In the special case of \( m = 2 \), we have a new combinatorial interpretation of the semi-Pell sequence and connections to other objects.

We enumerate partitions of the set $\{1,\dots,n\}$ according to occurrences of isolated successions, that is, integer strings $a,a+1,\dots,b$ in a block when neither $a-1$ nor $b+1$ lies in the same block. Our results include explicit formulas and generating functions for the number of partitions containing isolated successions of a given length. We also consider a corresponding analog of the associated Stirling numbers of the second kind.

We study new classes of overpartitions of numbers based on the properties of non-overlined parts. Several combinatorial identities are established by means of generating functions and bijective proofs. We show that our enumeration function satisfies a pair of infinite Ramanujantype congruences modulo 3. Lastly, by conditioning on the overlined parts of overpartitions,we give a seemingly new identity between the number of overpartitions and a certain class of ordinary partition functions. A bijective proof for this theorem also includes a partial answer to a previous request for a bijection on partitions doubly restricted by divisibility and frequency.

In this paper, we enumerate four new families of compositions whose members satisfy certain conditions on the sizes of the big (i.e., > 1) parts and/or lengths of the 1-strings.In particular, we consider various classes of compositions whose members do not contain two consecutive big parts.We make use of combinatorial arguments, mainly direct enumeration and bijections, in determining the cardinalities of these classes.As a consequence of our results, new combinatorial interpretations in terms of restricted compositions are obtained for some well-known sequences, including the Padovan, Narayana and m-step Fibonacci sequences.

A partition of a positive integer is full k -complete if it contains p ( t ) representations of every positive integer t, 1 ≤ t ≤ k , where p ( n ) is the number of all partitions of n . We study the enumeration properties of such partitions in this paper. We also discuss a further type of completeness defined by containment of perfect partitions.

Gutman index of a connected graph is a degree-distance-based topological index. In extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. In this paper, we compute the exact formulae of the Gutman indices for the four sum graphs (S-sum, R-sum, Q-sum, and T-sum) in the terms of various indices of their factor graphs, where sum graphs are obtained under the subdivision operations and Cartesian products of graphs. We also provide specific examples of our results and draw a comparison with previously known bounds for the four sum graphs.

A perfect composition of a positive integer is one whose sequence of parts contains one composition of every smaller positive integer. We obtain explicit formulas based on the observation that perfect compositions are identical with certain restricted permutations of perfect partitions. We also consider n-color perfect compositions.

George Andrews recently proved a new identity between the cardinalities of the set of Semi-Fibonacci partitions and the set of partitions into powers of two with all parts appearing an odd number of times. This paper extends the identity to the set of Semi-m-Fibonacci partitions of n and the set of partitions of n into powers of m in which all parts appear with multiplicity not divisible by m.

We prove a combinatorial identity between two classes of inverse-conjugate compositions, that is, integer compositions whose conjugates are given by a reversal of their sequences of parts. These are the set of inverse-conjugate compositions of 2n+3 without 2's, and the set of inverse-conjugate compositions of 2n-1 with parts not exceeding 3. Both sets are enumerated by 2F(n), where F-n is the nth Fibonacci number.

We consider certain classes of compositions of numbers based on the recently introduced extension of conjugation to higher orders. We use generating functions and combinatorial identities to provide enumeration results for compositions possessing conjugates of a given order. Working under some popular themes in the theory, we show that results for these compositions specialize to standard results in a natural way. We also give a generalization of MacMahon’s identities for inverse-conjugate compositions and discuss inverse-reciprocal compositions.

In analogy with the semi-Fibonacci partitions studied recently by Andrews, we define semi-Pell compositions and semi-$m$-Pell compositions. We find that these are in bijection with certain weakly unimodal $m$-ary compositions. We give generating functions, bijective proofs, and a number of unexpected congruences for these objects.