Can, Mahir Bilen A representation on the labeled rooted forests. (English) Zbl 1395.05142 Commun. Algebra 46, No. 10, 4273-4291 (2018). Summary: We consider the conjugation action of symmetric group on the semigroup of all partial functions and develop a machinery to investigate character formulas and multiplicities. By interpreting these objects in terms of labeled rooted forests, we give a characterization of the labeled rooted trees whose \(S_n\) orbit afford the sign representation. Applications to rook theory are offered. Cited in 1 Document MSC: 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C05 Trees 05E10 Combinatorial aspects of representation theory 20C30 Representations of finite symmetric groups 16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras) Keywords:labeled rooted trees; nilpotent partial transformations; plethysm; symmetric group PDF BibTeX XML Cite \textit{M. B. Can}, Commun. Algebra 46, No. 10, 4273--4291 (2018; Zbl 1395.05142) Full Text: DOI arXiv References: [1] Aker, K.; Can, M. B., From parking functions to Gelfand pairs, Proc. Amer. Math. Soc., 140, 4, 1113-1124, (2012) · Zbl 1254.20011 [2] Butler, F., Rook theory and cycle-counting permutation statistics, Adv. Appl. Math., 33, 4, 655-675, (2004) · Zbl 1056.05001 [3] Butler, F.; Can, M. B.; Haglund, J.; Remmel, J., Rook Theory, (2015) [4] Comtet, L., Advanced Combinatorics, (1974) [5] Doran, W. F. IV, Electron. J. Combin., 4, 1, (1997) [6] Ganyushkin, O.; Mazorchuk, V., Classical Finite Transformation Semigroups, Vol. 9 of Algebra and Applications, (2009), Springer-Verlag London, Ltd. An introduction, London · Zbl 1166.20056 [7] Garsia, A. M.; Haiman, M., A remarkable q,t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin., 5, 3, 191-244, (1996) · Zbl 0853.05008 [8] Garsia, A. M.; Remmel, J. B., Q-counting rook configurations and a formula of Frobenius, J. Combin. Theory Ser. A, 41, 2, 246-275, (1986) · Zbl 0598.05007 [9] Gessel, I. M.; Stanton, D., q-Series and Partitions (Minneapolis, MN, 1988), Vol. 18 of IMA Vol, Generalized rook polynomials and orthogonal polynomials, 159-176, (1988), Springer, Math. Appl.. New York [10] Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics, (1994), Addison-Wesley Publishing Company, Reading, MA · Zbl 0836.00001 [11] Haglund, J.; Haiman, M.; Loehr, N.; Remmel, J. B.; Ulyanov, A., A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J., 126, 2, 195-232, (2005) · Zbl 1069.05077 [12] Haglund, J., Rook theory and hypergeometric series, Adv. Appl. Math., 17, 4, 408-459, (1996) · Zbl 0865.05002 [13] Haglund, J., The q,t-Catalan numbers and the space of diagonal harmonics, Vol. 41 of University Lecture Series, (2008), American Mathematical Society, Providence, RI · Zbl 1142.05074 [14] Haiman, M.; Jerison, D.; Lusting, G.; Masur, B.; Mrowka, T.; Schmid, W.; Stanley, R.; Yau, S.-T., Combinatorics, symmetric functions, and Hilbert schemes, 39-111, (2003), Int, Somerville, MA [15] Howe, R., (GLn, GL_m)-duality and symmetric plethysm, Proc. Indian Acad. Sci. Math. Sci., 97, 1-3, 85-109, (1987) [16] Kaplansky, I.; Riordan, J., The problem of the rooks and its applications, Duke Math. J., 13, 259-268, (1946) · Zbl 0060.02903 [17] Knuth, D. E., The Art of Computer Programming. Vol. 3, (1998), Addison-Wesley, Reading, MA [18] Laradji, A.; Umar, A., On the number of nilpotents in the partial symmetric semigroup, Commun. Algebra, 32, 8, 3017-3023, (2004) · Zbl 1069.20061 [19] Littlewood, D. E., Invariant theory, tensors and group characters, Philos. Trans. Roy. Soc. London. Ser. A, 239, 305-365, (1944) · Zbl 0060.04402 [20] Loehr, N. A.; Remmel, J. B., A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin., 33, 2, 163-198, (2011) · Zbl 1229.05275 [21] Macdonald, I. G., Symmetric Functions and Hall Polynomials, (1995) · Zbl 0899.05068 [22] Putcha, M. S., Linear Algebraic Monoids, Vol. 133 of London Mathematical Society Lecture Note Series, (1988), Cambridge University Press, Cambridge · Zbl 0647.20066 [23] Renner, L. E., Linear Algebraic Monoids, Vol, 134 of Encyclopaedia of Mathematical Sciences, (2005), Springer-Verlag, Berlin [24] Sagan, B. E., Partially Ordered Sets with Hooklengths - An Algorithmic Approach, (1979), ProQuest LLC, Ann Arbor, MI [25] Stanley, R. P. Stembridge, On immanants of Jacobi-trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A, 62, 2, 261-279, (1993) · Zbl 0772.05097 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.