Enumeration of Even Dimensional Partitions modulo 4
Aditya Khanna

TL;DR
This paper extends the enumeration of partitions with odd dimensions to those with dimensions congruent to 2 modulo 4, providing explicit formulas and recursive methods for their enumeration.
Contribution
It introduces a new enumeration for partitions with dimensions ≡ 2 mod 4 using 2-core tower theory, extending previous work on odd dimensions.
Findings
Explicit formulas for a_2(n) when n has no consecutive 1s in binary
Recursive formula for computing a_2(n) for all n
Extension of enumeration from odd dimensions to dimensions ≡ 2 mod 4
Abstract
The number of standard Young tableaux possible of shape corresponding to a partition is called the dimension of the partition and is denoted by . Partitions with odd dimensions were enumerated by McKay and were further characterized by Macdonald using the theory of 2-core towers. We use the same theory to extend the results to partitions of with dimensions congruent to 2 modulo 4 which are enumerated by . We provide explicit results for when has no consecutive 1s in its binary expansion and give a recursive formula to compute for all .
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Algebraic structures and combinatorial models
