4 citations · 5 across the 14 of their papers we have counts for
14 papers
Beyond Bass Collapse: New Irregular Edge-Space Invariants in Ihara Theory
Hartosh Singh Bal
Let \(G\) be a finite simple graph and let \(T\) be its Hashimoto operator on the directed-edge space. We show that edge reversal induces a canonical symmetric/antisymmetric splitt…
The Antisymmetric Line Graph
Hartosh Singh Bal
Let be a finite simple graph with oriented incidence matrix . The signed graph on edge set with adjacency matrix \[ A_{\mathcal A(G)}=D^{\mathsf T}D-2I \] is classica…
Partition Frequency Moments: Modularity and Congruences
Hartosh Singh Bal
We study frequency moments of partition statistics arising from Euler products via a transform that expresses the moment generating functions as…
Constancy of an Infinite Cyclotomic Product via Ramanujan Sums
Hartosh Singh Bal
We show that the infinite product defined by \[ P(z) = -\prod_{n=1}^{\infty} (Φ_n(z))^{-1/n}, \] where \( Φ_n(z) \) is the \( n \)-th cyclotomic polynomial, is constant inside the…
Dold-Gauss Congruences, Norm Descent, and Rational Rigidity
Hartosh Singh Bal
We develop a Witt--Hadamard calculus for Euler products that unifies the classical Gauss congruences with their modern refinement, the Dold congruences. Within this framework we pr…
An easy proof of Ramanujan's famous congruences
Hartosh Singh Bal, Gaurav Bhatnagar
We present a proof of Ramanujan's congruences The proof only requires a limiting case of Jacobi's triple product…
Persistent Quantum Memory in Iterated Lifts
Hartosh Singh Bal
We study quantum coherence in continuous-time quantum walks on perfect graphs generated by the symmetric lift , a canonical, unweighted, undirected constructio…
Perfecting the Line Graph
Hartosh Singh Bal
We study the doubled edge-stage lift \[ \HL'_2(G)=L(G\otimes K_2), \] the line graph of the canonical bipartite double cover of a graph \(G\). The natural involution \((u,v)\leftri…
Completely Additive Height Functions: Profile Laws, Matula Bounds, and Inverse Growth
Hartosh Singh Bal
The height (H(n)) of an integer (n) is classically the number of iterations of Euler's totient function required to reach (1). H. N. Shapiro showed that a modification of this func…
Lognormal Degree Distribution in the Partition Graphs
Hartosh Singh Bal
We demonstrate a method for listing all ordinary partitions of n as binary words of length (n-1). The resulting family imbued with the hamming distance yields subgraphs of the Hamm…
Stanley--Elder--Fine theorems for colored partitions
Hartosh Singh Bal, Gaurav Bhatnagar
We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored parti…
Glaisher's divisors and infinite products
Hartosh Singh Bal, Gaurav Bhatnagar
Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function . In 1885, J.W. Glaisher considered seven divisor sums closely rel…
The Partition-Frequency Enumeration Matrix
Hartosh Singh Bal, Gaurav Bhatnagar
We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Part…
Prime number conjectures from the Shapiro class structure
Hartosh Singh Bal, Gaurav Bhatnagar
The height of , introduced by Pillai in 1929, is the smallest positive integer such that the th iterate of Euler's totient function at is . H. N. Shapiro (1…