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A representation on the labeled rooted forests. (English) Zbl 1395.05142
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.

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)
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[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 exposĂ© 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
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