Order, Structure and Reality: Hierarchical Shell Structure of Integer Partitions
Abstract
I construct a self-unfolding combinatorial structure from a single integer seed nc0 ∈ Z≥0: a finite, bilaterally symmetric double-cone hierarchy whose total cell count equals the partition function p(nc0). The central contribution is the construction of a regular generative structure for integer partitions, a single object that deterministically generates all partitions of nc0 through its intrinsic geometry. The framework connects partition theory with mixed-radix systems, conical geometry, and signed partitions via a tunable non-decreasing constraint. For standard partitions, a digital root invariance theorem shows that every path label is congruent to the seed modulo 9, confining the generated universe to one of nine equivalence classes ED(nc0 ).
Related
- Order, Structure and Reality: A Unified Hierarchical Framework
- Excedances and Deficiencies of Integer Partitions
- Intersecting integer partitions: star bounds and counterexamples at every scale
- Estimating the asymptotics of integer partitions in intermediate dimensions (d = 3,4,5,6)
- Integer-Quantized Hierarchical Matrices with Adaptive Bit Allocation
BibTeX
@misc{akbarpour2026order,
title = {{Order, Structure and Reality: Hierarchical Shell Structure of Integer Partitions}},
author = {Masoud Akbarpour},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05776192v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Order, Structure and Reality: Hierarchical Shell Structure of Integer Partitions
AU - Masoud Akbarpour
PY - 2026
DA - 2026/10/02
PB - HAL
UR - https://hal.science/hal-05776192v1
AB - I construct a self-unfolding combinatorial structure from a single integer seed nc0 ∈ Z≥0: a finite, bilaterally symmetric double-cone hierarchy whose total cell count equals the partition function p(nc0). The central contribution is the construction of a regular generative structure for integer partitions, a single object that deterministically generates all partitions of nc0 through its intrinsic geometry. The framework connects partition theory with mixed-radix systems, conical geometry, and signed partitions via a tunable non-decreasing constraint. For standard partitions, a digital root invariance theorem shows that every path label is congruent to the seed modulo 9, confining the generated universe to one of nine equivalence classes ED(nc0 ).
ER -