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Hartosh Singh Bal

Publications and source records attributed to Hartosh Singh Bal.

14 recordsLinked to original sources
arXiv2026

Completely Additive Height Functions: Profile Laws, Matula Bounds, and Inverse Growth

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 function is completely additive. We study completely additive height functions with finite prime fibers. Their prime-height profile (π_k) determines the height multiplicities (N_k) through the weighted-multipartition identity (\sum_k N_k q^k=\prod_j(1-q^j)^{-π_j}), and conversely every profile containing infinitely many primes is realizable. We introduce iteratively defined heights encompassing Shapiro-type totient heights and the Matula height. For the Matula height, we give purely number-theoretic proofs of the classical Gutman-Ivi'c extremal bounds, thereby answering their question whether the maximal bound can be derived without recourse to the rooted-tree interpretation. Using Meinardus' theorem in its full form, we prove a conditional inverse-growth law: if (Π_k\sim Ck^α), then (\log N_k\sim C_2 k^{α/(α+1)}), with an explicit constant. We also derive average-order results for a canonical sequential realization and report computations for the Shapiro height beyond the polynomial regime.

math.CO↗
arXiv2026

Dold-Gauss Congruences, Norm Descent, and Rational Rigidity

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 prove \emph{norm descent}: Dold congruences are functorial under finite extensions and preserved by prime--ideal norms $N_{K/\mathbb{Q}}$, yielding integer ghosts from algebraic ones. We extend the theory from $\mathbb{Z}$ to Dedekind domains, and show that integrality is stable under both Hadamard and Witt products. Two rigidity theorems lie at the core: a \emph{cyclotomic residues theorem}, asserting that if the logarithmic derivative has only cyclotomic poles then integrality forces rationality; and a stronger \emph{Dold$^{+}$ rigidity theorem}, showing that any algebraic series satisfying refined Dold congruences is necessarily rational. These results sharpen the Gauss--Dold picture: ordinary congruences enforce integrality, while the strengthened form collapses algebraic cases to rational ones. Applications include prime--ideal ladders in number fields and exact product laws for dynamical zeta functions, illustrated for subshifts of finite type and circle doubling.

math.NT↗
arXiv2026

Beyond Bass Collapse: New Irregular Edge-Space Invariants in Ihara Theory

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 splitting under which \(T\) acquires an explicit \(2\times 2\) block form. The diagonal blocks are \(\tfrac12 L(G)\) and \(-\tfrac12 A(G)\), where \(L(G)\) is the line-graph adjacency and \(A(G)\) is the antisymmetric line-graph adjacency, while the off-diagonal block is the mixed incidence product \(M=|D|^\top D\). This identifies the ordinary and antisymmetric line-graph sectors as the two canonical diagonal sectors of Hashimoto theory and isolates a mixed sector linking them. A Schur-complement argument then gives a factorization \[ \det(I-wT)=\det\!\bigl(I-\tfrac w2 L(G)\bigr)\,C_G(w), \] where \(C_G(w)\) is an explicit correction determinant built from the antisymmetric and mixed sectors. We show that the trivial roots \(w=\pm1\) localize on canonical edge subspaces, and that for line-graph-cospectral pairs all remaining Ihara separation is forced into the correction sector. Although the raw mixed block \(M\) depends on edge orientation, its natural gauge-invariant shadows, including \(MM^\top\), \(M^\top M\), and \(M^\top L^kM\), define a canonical matrix package attached to the graph. In the regular case these collapse to adjacency-side data, but in the irregular case they need not. As an application, we exhibit irregular non-isomorphic graphs that are adjacency-cospectral and line-graph-cospectral yet are separated by the correction sector, and we find further examples where the gauge-invariant mixed shadows separate even when the scalar Ihara polynomial does not. This isolates new irregular edge-space invariants in Hashimoto--Ihara theory.

math.CO↗
arXiv2026

Perfecting the Line Graph

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 \((u,v)\leftrightarrow(v,u)\) has quotient isomorphic to \(L(G)\), and induces a sector decomposition \[ \Spec(\HL'_2(G))=\Spec(L(G))\cup\Spec(\mathcal A(G)), \] where \(\mathcal A(G)\) is a canonical signed refinement of the line graph. Thus the construction retains substantial edge-space information through its quotient and antisymmetric sector. For every input graph, \(\HL'_2(G)\) is perfect, claw-free, and box-perfect. In the regular case we give an explicit spectral formula, together with quantitative control of the second eigenvalue and spectral gap for non-bipartite input. Explicit families, including the complete-graph lifts and the Paley lifts, illustrate the theory; in particular, the Paley lifts furnish an explicit family of regular perfect graphs with controlled adjacency spectrum and spectral gap. The construction may be viewed both intrinsically, via ordered-edge adjacency by one-coordinate agreement, and extrinsically, as the line graph of the canonical double cover. The first viewpoint emphasizes the edge-stage nature of the lift, while the second supplies the structural proofs used here.

math.CO↗
arXiv2026

The Antisymmetric Line Graph

Let $G$ be a finite simple graph with oriented incidence matrix $D$. The signed graph on edge set $E(G)$ with adjacency matrix \[ A_{\mathcal A(G)}=D^{\mathsf T}D-2I \] is classical in the signed-line-graph literature. In this paper we study its canonical switching class as a source of invariants of the underlying unsigned graph. We prove that the switching class of $\mathcal A(G)$ determines $G$ up to isomorphism modulo isolated vertices, and we relate the frustration index $\ell(\mathcal A(G))$ to classical bipartization parameters. In particular, we show \[ \operatorname{def}(G)\le \ell(\mathcal A(G))\le (Δ(G)-1)\operatorname{def}(G), \] and, for cubic graphs, \[ \ell(\mathcal A(G))=2\,\operatorname{oct}(G). \] We then prove the exact optimization identity \[ \ell(\mathcal A(G)) = \frac14\sum_{v\in V(G)} d(v)^2 -\frac14\max_{x\in\{\pm1\}^{E(G)}}\|Dx\|^2, \] so $\ell(\mathcal A(G))$ is exactly a Boolean edge-space Laplacian optimization problem. This yields a spectral lower bound in terms of the largest Laplacian eigenvalue, a cubic spectral lower bound on odd cycle transversal, and explicit family-level comparisons showing that the spectral and defect bounds govern different regimes: on odd cycles the spectral bound is asymptotically vacuous, while on complete multipartite graphs it already captures exactly $3/4$ of the true value of $\ell(\mathcal A(G))$. Thus the paper uses a classical signed line graph in a new way: as a source of combinatorial invariants of ordinary graphs, especially through frustration and odd-cycle-transversal phenomena.

math.CO↗
arXiv2026

Partition Frequency Moments: Modularity and Congruences

We study frequency moments of partition statistics arising from Euler products $A(q)=\prod_{r\ge1}(1-q^r)^{-c(r)}$ via a transform that expresses the moment generating functions as $B(q)$ times explicit divisor--sum series determined by $c(r)$. When $A(q)$ is modular (typically an $η$--quotient), this yields (quasi)modular forms whose coefficients can be projected to arithmetic progressions and certified modulo primes by a Sturm bound, giving an effective pipeline for detecting and proving Ramanujan--type congruences for frequency moments. For ordinary partitions we recover and certify several congruences for odd moments in nonzero residue classes (e.g.\ $M_3(7n+5)\equiv 0\pmod7$ and $M_3(11n+6)\equiv 0\pmod{11}$). As a second input, we apply the same pipeline to overpartitions and certify a family of zero--class congruences $M_m^{\overline{\ }}(\ell n)\equiv 0\pmod{\ell}$ (including $m=5,7,11,13$), exhibiting a sharp contrast with the ordinary partition case: no nonzero residue--class congruences are observed for overpartition moments in our scan range. We also demonstrate that filtering the statistic via the Glaisher--character dictionary can itself create new Ramanujan--type progressions, e.g.\ a quadratic twist yields the certified congruence $\widehat{M}^{χ_5}_3(5n+4)\equiv 0\pmod{5}$.

math.NT↗
arXiv2026

The Partition-Frequency Enumeration Matrix

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 Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan's $τ$ function, sums of squares and triangular numbers, and for $ζ(2n)$, where $n$ is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer and others. As one application, we embed Ramanujan's famous congruences $p(5n+4)\equiv 0$ (mod $5)$ and $τ(5n+5)\equiv 0$ (mod $5)$ into an infinite family of such congruences.

math.NT↗
arXiv2026

Constancy of an Infinite Cyclotomic Product via Ramanujan Sums

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 unit disk. The proof translates a result of Ramanujan on Ramanujan sums, equivalent to the prime number theorem, to the setting of infinite products. We also show that similar identities proved by Ramanujan lead to additional results on infinite cyclotomic products.

math.NT↗
arXiv2025

Persistent Quantum Memory in Iterated Lifts

We study quantum coherence in continuous-time quantum walks on perfect graphs generated by the symmetric lift ${\mathrm{HL}}'_2(G)$, a canonical, unweighted, undirected construction defined as the line graph of a bipartite double cover of $G$. This lift acts as both a coherence-preserving and coherence-inducing transformation: it preserves and scales structured quantum interference in highly symmetric base graphs, and induces sustained coherence in random or weakly structured ones. In small graphs such as $K_4$, $K_5$, and the Petersen graph, where quantum walks exhibit sharp revivals and high return probability, repeated $\mathrm{HL}'_2$ lifting produces towers of perfect graphs with thousands to tens of thousands of vertices that retain periodic or quasi-periodic coherence. When applied to random regular or Erdős--Rényi graphs with flat or decaying return behavior, the lift introduces structured interference and significant amplification of mean and peak return probabilities. To quantify these effects, we evaluate standard coherence metrics from quantum resource theory, including inverse participation ratio (IPR), purity, relative entropy of coherence, and the logarithmic coherence number. These measures confirm that $\mathrm{HL}'_2$ lifting delocalizes eigenstates, increases coherence entropy, and expands the basis support of quantum states. These results demonstrate that $\mathrm{HL}'_2$ is a scalable and structurally grounded mechanism for organizing quantum interference, and introduce a new family of perfect graphs that support long-time quantum coherence without spectral tuning or engineered weights.

quant-ph↗
arXiv2023

Lognormal Degree Distribution in the Partition Graphs

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 Hamming Graphs. The existence of a 2-Gray Code for ordinary partitions follows from the fact that the graph (with the all 0s partition omitted) is 2-connected. However, the graphs fail to be hamiltonian for ordinary partitions when n > 7, ruling out the possibility of a Gray code for all such flip graphs. We further investigate the degree distribution of the graph for n, and provide computational evidence that this is a long-tailed lognormal distribution. This conjecture connects to a closely related, and much older, question of the distribution of the number of parts of a partition and the same evidence suggests that this distribution is also lognormal for large n. These methods extend to higher dimensional partitions of n which can be then written as words of length (n-1) on d + 1 letters. The resulting graphs are connected, proving that d-dimensional partitions allow a 3-Gray code.

math.CO↗
arXiv2022

Glaisher's divisors and infinite products

Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function $σ(n)$. In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a calculus to associate a generating function with each of these divisor sums. This yields analogues of Ramanujan's recurrence relation for several partition-theoretic functions as well as $r_k(n)$ and $t_k(n)$, functions counting the number of ways of writing a number as a sum of squares (respectively, triangular) numbers. As by-products of this association, we obtain several convolutions, recurrences and congruences for divisor functions. We give alternate proofs of two classical theorems, one due to Legendre and the other -- Ramanujan's congruence $p(5n+4) \equiv 0 \pmod 5$.

math.NT↗
arXiv2021

Stanley--Elder--Fine theorems for colored partitions

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 partitions (or prefabs). More specifically, we give analogous results for $b$-colored partitions, where each part occurs in $b$ colors; for $b$-colored partitions with odd parts (or distinct parts); for partitions where the part $k$ comes in $k$ colors; and, overpartitions.

math.CO↗
arXiv2020

Prime number conjectures from the Shapiro class structure

The height $H(n)$ of $n$, introduced by Pillai in 1929, is the smallest positive integer $i$ such that the $i$th iterate of Euler's totient function at $n$ is $1$. H. N. Shapiro (1943) studied the structure of the set of all numbers at a height. We state a formula for the height function due to Shapiro and use it to list steps to generate numbers at any height. This turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. We present some theoretical and computational evidence to show that $H$ and its relatives are closely related to the important functions of number theory, namely $π(n)$ and the $n$th prime $p_n$. We conjecture formulas for $π(n)$ and $p_n$ in terms of the height function.

math.NT↗