ALGEBRAIC COMBINATORICS

no. 5
Loop-augmented Forests and a Variant of Foulkes’s Conjecture
Can, Mahir Bilen1; Remmel, Jeff2 †
1 Tulane University Mathematics Department New Orleans, USA
2 UC San Diego Mathematics Department La Jolla, USA
Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 573-601.

A loop-augmented forest is a labeled rooted forest with loops on some of its roots. By exploiting an interplay between nilpotent partial functions and labeled rooted forests, we investigate the permutation action of the symmetric group on loop-augmented forests. Furthermore, we describe an extension of Foulkes’s conjecture and prove a special case. Among other important outcomes of our analysis are a complete description of the stabilizer subgroup of an idempotent in the semigroup of partial transformations and a generalization of the (Knuth–Sagan) hook length formula.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.20
Classification: 20C30, 05E10, 16W22
Keywords: Labeled rooted forests, symmetric group, plethysm.
Can, Mahir Bilen 1; Remmel, Jeff 2

1 Tulane University Mathematics Department New Orleans, USA
2 UC San Diego Mathematics Department La Jolla, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Can, Mahir Bilen; Remmel, Jeff. Loop-augmented Forests and a Variant of Foulkes’s Conjecture. Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 573-601. doi : 10.5802/alco.20. https://alco.centre-mersenne.org/articles/10.5802/alco.20/

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