Loop-augmented Forests and a Variant of Foulkes’s Conjecture
Algebraic Combinatorics, Volume 1 (2018) no. 5, p. 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 : 2017-08-09
Revised : 2018-03-01
Accepted : 2018-05-07
Published online : 2018-11-30
DOI : https://doi.org/10.5802/alco.20
Classification:  20C30,  05E10,  16W22
Keywords: Labeled rooted forests, symmetric group, plethysm.
@article{ALCO_2018__1_5_573_0,
     author = {Can, Mahir Bilen and Remmel, Jeff},
     title = {Loop-augmented Forests and a Variant of Foulkes's Conjecture},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {1},
     number = {5},
     year = {2018},
     pages = {573-601},
     doi = {10.5802/alco.20},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_5_573_0}
}
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/item/ALCO_2018__1_5_573_0/

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