On the number of direct-sum decompositions of a finite vector space
David Ellerman · Open Journal of Mathematical Sciences · 2026
The theory of q-analogs develops combinatorial formulas for finite vector spaces over a finite field with q elements–in analogy with formulas for finite sets (the limiting case q = 1). A direct-sum decomposition of a finite vector space is the vector-space analogue of a set partition. This paper uses elementary counting methods to derive direct formulas for the number of direct-sum decompositions (DSDs) that play the role of the Stirling and Bell numbers for set partitions. In particular, we give a signature-based counting formula for DSDs and recover the standard set-partition formulas in the limit q → 1. We also develop new companion formulas that enumerate DSDs with m blocks in an n-dimensional vector space over GF(q) such that a specified nonzero vector lies in one of the blocks, together with the corresponding totals over all numbers of blocks. Initial computations are included for the case q = 2, with hand-checkable low-dimensional examples, internal consistency checks, and applications to the pedagogical model of quantum mechanics over ℤ2 (QM/Sets). Four related sequences for q = 2 are recorded in the On-Line Encyclopedia of Integer Sequences