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The algebraic combinatorics of snakes. (English) Zbl 1246.05164
Summary: Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type $$B$$ noncommutative symmetric functions, noncommutative supersymmetric functions and colored free quasi-symmetric functions.

MSC:
 05E05 Symmetric functions and generalizations 05E15 Combinatorial aspects of groups and algebras (MSC2010) 20F55 Reflection and Coxeter groups (group-theoretic aspects)
OEIS
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References:
 [1] André, D., Sur LES permutations alternées, J. math. pures appl., 7, 167-184, (1881) · JFM 13.0152.02 [2] Arnolʼd, V.I., The calculus of snakes and the combinatorics of Bernoulli, Euler, and Springer numbers for Coxeter groups, Russian math. surveys, 47, 1-51, (1992) · Zbl 0791.05001 [3] Björner, A., Orderings of Coxeter groups, (), 175-195 [4] C.-O. Chow, Noncommutative symmetric functions of type B, Thesis, MIT, 2001. [5] Duchamp, G.; Hivert, F.; Thibon, J.-Y., Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Internat. J. algebra comput., 12, 671-717, (2002) · Zbl 1027.05107 [6] Ehrenborg, R.; Readdy, M., Sheffer posets and r-signed permutations, Ann. sci. math. Québec, 19, 173-196, (1995) · Zbl 0843.06003 [7] Gelfand, I.M.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V.S.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. math., 112, 218-348, (1995) · Zbl 0831.05063 [8] Hivert, F.; Novelli, J.-C.; Thibon, J.-Y., The algebra of binary search trees, Theoret. comput. sci., 339, 129-165, (2005) · Zbl 1072.05052 [9] Hivert, F.; Novelli, J.-C.; Thibon, J.-Y., Trees, functional equations and combinatorial Hopf algebras, European J. combin., 29, 1, 1682-1695, (2008) · Zbl 1227.05272 [10] Hoffman, M., Derivative polynomials, Euler polynomials, and associated integer sequences, Electron. J. combin., 6, #R21, (1999) · Zbl 0933.11005 [11] Josuat-Vergès, M., Enumeration of snakes and cycle-alternating permutations, preprint · Zbl 1305.05011 [12] Krob, D.; Leclerc, B.; Thibon, J.-Y., Noncommutative symmetric functions II: transformations of alphabets, Internat. J. algebra comput., 7, 181-264, (1997) · Zbl 0907.05055 [13] Mantaci, R.; Reutenauer, C., A generalization of solomonʼs descent algebra for hyperoctahedral groups and wreath products, Comm. algebra, 23, 27-56, (1995) · Zbl 0836.20010 [14] Novelli, J.-C.; Thibon, J.-Y., Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions, Discrete math., 30, 3584-3606, (2010) · Zbl 1231.05278 [15] Novelli, J.-C.; Thibon, J.-Y., Superization and $$(q, t)$$-specialization in combinatorial Hopf algebras, Electron. J. combin., 16, 2, R21, (2009) · Zbl 1230.05281 [16] Reutenauer, C., Free Lie algebras, (1993), Oxford University Press · Zbl 0798.17001 [17] Sloane, N.J.A., The on-line encyclopedia of integer sequences, (electronic) · Zbl 1274.11001 [18] Springer, T.A., Remarks on a combinatorial problem, Nieuw arch. wiskd., 19, 30-36, (1971) · Zbl 0224.05002
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