Browsing by Author "Eyyunni, Pramod"

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Bressoud–Subbarao Type Weighted Partition Identities for a Generalized Divisor Function
(Springer, 2023-04) Eyyunni, Pramod
In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity of Bressoud and Subbarao starting from a q-series identity of Ramanujan. In the present paper, we revisit the combinatorial arguments of Bressoud and Subbarao, and derive a more general weighted partition identity. Furthermore, with the help of a fractional differential operator, we establish a few more Bressoud– Subbarao type weighted partition identities beginning from an identity of Andrews, Garvan and Liang. We also found a one-variable generalization of an identity of Uchimura related to Bell polynomials.
Combinatorial properties of sparsely totient numbers
(ARXIV, 2020) Eyyunni, Pramod
Let N1(m) = maxfn: (n) mg and N1 = fN1(m) : m 2 (N)g where (n) denotes the Euler's totient function. Masser and Shiu [3] call the elements of N1 as `sparsely totient num- bers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all su ciently large sparsely totient numbers and a non-squarefree integer divides in nitely many sparsely totient numbers. Next, we construct explicit in nite families of sparsely totient numbers and describe their relationship with the distribution of consecutive primes. We also study the sparseness of N1 and prove that it is multiplicatively piecewise syndetic but not additively piecewise syndetic. Finally, we investigate arithmetic/geometric progressions and other additive and multiplicative patterns like fx; y; x + yg; fx; y; xyg; fx + y; xyg and their generalizations in the sparsely totient numbers.
Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities
(Springer, 2022-01) Eyyunni, Pramod
Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From their generalized identity, they were able to derive the last three of these q-series identities but did not establish the first two. In the present article, we derive a one-variable generalization of the main identity of Dixit and the third author from whichwe successfully deduce all the five q-series identities of Ramanujan. In addition to this, we also establish a few interesting weighted partition identities from our generalized identity. In the mid 1980s, Bressoud and Subbarao found an interesting identity connecting the generalized divisor function with a weighted partition function, which they proved by means of a purely combinatorial argument. Quite surprisingly, we found an analytic proof for a generalization of the identity of Bressoud and Subbarao, starting from the fourth identity of the aforementioned five q-series identities of Ramanujan.
An inequality between finite analogues of rank and Crank moments
(ARXIV, 2019-08) Eyyunni, Pramod
The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite ana- logues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews’ famous identity for smallest parts function. In the same paper, they also con- jectured an inequality between finite analogues of rank and crank moments, analogous to Garvan’s conjecture. In the present paper, we give a proof of this conjecture
Minimal excludant over partitions into distinct parts
(ARXIV, 2022) Eyyunni, Pramod
The average size of the “smallest gap” of a partition was studied by Grabner and Knopfmacher in 2006. Recently, Andrews and Newman, motivated by the work of Fraenkel and Peled, studied the concept of the “smallest gap” under the name “minimal excludant” of a partition and rediscovered a result of Grabner and Knopfmacher. In the present paper, we study the sum of the minimal excludants over partitions into distinct parts, and interestingly we observe that it has a nice connection with Ramanujan’s function (q). As an application, we derive a stronger version of a result of Uncu.
A new generalization of the minimal excludant arising from an analogue of Franklin's identity
(Elsevier, 2023-05) Eyyunni, Pramod
Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly j different multiples of r, for a positive integer r. We prove an analogue of Franklin's identity by studying the number of partitions with j multiples of r in total and in the process, discover a natural generalization of the minimal excludant (mex) which we call the r-chain mex. Further, we derive the generating function for , the sum of r-chain mex taken over all partitions of n, thereby deducing a combinatorial identity for , which neatly generalizes the result of Andrews and Newman for , the sum of mex over all partitions of n.
On the local structure of the set of values of Euler's φ function
(ARXIV, 2021-03) Eyyunni, Pramod
Assuming the validity of Dickson's conjecture, we show that the set V of values of the Euler's totient function φ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of proving unconditionally that this set V has a positive upper Banach density.
On thin sum-product bases
(Springer, 2021) Eyyunni, Pramod
Additive bases, and less importantly multiplicative bases, have been ex- tensively studied for several centuries. More recently, expanding polynomi- als (of course, with more than one variable) have been considered with a view to studying the expansion of nite sets under these polynomials. If f 2Z[x1;x2; : : : ;xd] and A is contained in a given subset R of a commutative ring, then let f(A;A;: : : ;A) (with k arguments) denote the set of all terms f(a1;a2; : : : ;ak) where the ai's belong to A. The polynomial f is called an expander if there exists >0 such that jf(A;: : : ;A)j>jAj1+ for any nite set A, where jBj denotes the cardinality of a nite set B. If R is nite, as for instance, if R=Fq or f1; : : : ;Ng, we need to restrict the above de nition by assuming that jRj"0. A more restrictive no- tion is of a covering polynomial which arises from the following question: is there a non trivial minimal size such that if A attains it, then f(A;A;: : : ;A) entirely covers R?
A refinement of a result of Andrews and Newman on the sum of minimal excludants
(Springer, 2023-06) Eyyunni, Pramod
In this article, we refine a result of Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number n equals the number of partitions of n into distinct parts with two colors. As a consequence, we find congruences modulo 4 and 8 for the functions appearing in this refinement. We also conjecture three further congruences for these functions. In addition, we also initiate the study of kth moments of minimal excludants. At the end, we also provide an alternate proof of a beautiful identity due to Hopkins, Sellers, and Stanton.
The second minimal excludant and mex sequences
(Rocky Mountain Mathematics Consortium, 2024-08) Eyyunni, Pramod
The minimal excludant of an integer partition, first studied prominently by Andrews and Newman from a combinatorial viewpoint, is the smallest positive integer missing from a partition. Several generalizations of this concept are being explored by mathematicians nowadays. We analogously consider the second minimal excludant of a partition and analyze its relationship with the minimal excludant. This leads us to the notion of a mex sequence and we derive two neat identities involving the number of partitions whose mex sequence has length at least r
Sparse subsets of the natural numbers and Euler’s totient function
(Springer, 2019-08) Eyyunni, Pramod
In this article, we investigate sparse subsets of the natural numbers and study the sparseness of some sets associated to the Euler’s totient function φ via the property of ‘Banach density’. These sets related to the totient function are defined as follows: V := φ(N) and Ni := {Ni (m) : m ∈ V} for i = 1, 2, 3, where N1(m) = max{x ∈ N: φ(x) ≤ m}, N2(m) = max(φ −1(m)) and N3(m) = min(φ −1(m)) for m ∈ V. Masser and Shiu (Pacific J. Math. 121(2) (1986) 407–426) called the elements of N1 as ‘sparsely totient numbers’ and constructed an infinite family of these numbers. Here we construct several infinite families of numbers in N2 \ N1 and an infinite family of composite numbers in N3. We also study (i) the ratio N2(m) N3(m) which is linked to the Carmichael’s conjecture, namely, |φ −1(m)| ≥ 2 for all m ∈ V, and (ii) arithmetic and geometric progressions in N2 and N3. Finally, using the above sets associated to the totient function, we generate an infinite class of subsets of N, each with asymptotic density zero and containing arbitrarily long arithmetic progressions.
Untrodden pathways in the theory of the restricted partition function p(n,N)
(Elsevier, 2021-05) Eyyunni, Pramod
We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for . The latter motivates us to extend the theory of the restricted partition function , namely, the number of partitions of n with largest parts less than or equal to N, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for our finite analogue of the spt-function, namely , we prove an inequality between the finite second rank and crank moments. The other results obtained include finite analogues of a recent identity of Garvan, an identity relating and , namely the finite analogues of the divisor and largest parts functions respectively, and a finite analogue of the Beck-Chern theorem.