Gauss Congruences and Triangular-Number Sign Patterns in Restricted Set Partitions
Year: 2026
Type: preprint
Abstract: We consider labeled set partitions in which each block size can occur at most once per color. Their signed and two-colored exponential generating functions give OEIS A185895 and A032312, respectively. A cyclic action proves the Gauss congruences for the entire colored, signed family, encompassing the congruence questions recorded for both sequences. For the signed sequence we establish a uniform comparison of adjacent block-rank masses, yielding its triangular-number sign pattern with an explicit nonvanishing bound. We also establish the same sign pattern for the square of its ordinary generating function. The last conclusion combines an explicit analytic estimate for all sufficiently large indices with directed-rounding interval certificates for the complementary finite range. These arguments separate general theorems from the finite computations needed for the ordinary-square assertion. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.
Author: Su, Lien-Hung
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223685374