Square-bounded partitions and Catalan numbers
Matthew Bennett
, Vyjayanthi Chari
, R.J. Dolbin3
and Nathan Manning4
3V. Chari
DOI: 10.1007/s10801-010-0260-6
Abstract
For each integer k\geq 1, we define an algorithm which associates to a partition whose maximal value is at most k a certain subset of all partitions. In the case when we begin with a partition λ which is square-bounded, i.e. λ =( λ 1\geq \cdot \cdot \cdot \geq λ k ) with λ 1= k and λ k =1, applying the algorithm \ell times gives rise to a set whose cardinality is either the Catalan number c \ell - k+1 (the self dual case) or twice that Catalan number. The algorithm defines a tree and we study the propagation of the tree, which is not in the isomorphism class of the usual Catalan tree. The algorithm can also be modified to produce a two-parameter family of sets and the resulting cardinalities of the sets are the ballot numbers. Finally, we give a conjecture on the rank of a particular module for the ring of symmetric functions in 2 \ell + m variables.
Pages: 1–18
Keywords: keywords partitions; Young diagrams; Catalan numbers; current algebras
Full Text: PDF
References
1. Aigner, M.: A Course in Enumeration. Springer, Berlin (2007)
2. Beck, J., Chari, V., Pressley, A.: An algebraic characterization of the affine canonical basis. Duke Math. J. 99(3), 455-487 (1999)
3. Bennett, M., Chari, V., Greenstein, J., Manning, N.: On homomorphisms between global Weyl modules. Preprint,
4. Chari, V., Greenstein, J.: Current algebras, highest weight categories and quivers. Adv. Math. 216(2), 811-840 (2007)
5. Chari, V., Greenstein, J.: A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras. Adv. Math. 220(4), 1193-1221 (2009)
6. Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras. Represent. Theory 5, 191-223 (2001) (electronic)
7. Cline, E.T., Parshall, B.J., Scott, L.L.: Finite-dimensional algebras and highest weight categories.
2. Beck, J., Chari, V., Pressley, A.: An algebraic characterization of the affine canonical basis. Duke Math. J. 99(3), 455-487 (1999)
3. Bennett, M., Chari, V., Greenstein, J., Manning, N.: On homomorphisms between global Weyl modules. Preprint,
4. Chari, V., Greenstein, J.: Current algebras, highest weight categories and quivers. Adv. Math. 216(2), 811-840 (2007)
5. Chari, V., Greenstein, J.: A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras. Adv. Math. 220(4), 1193-1221 (2009)
6. Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras. Represent. Theory 5, 191-223 (2001) (electronic)
7. Cline, E.T., Parshall, B.J., Scott, L.L.: Finite-dimensional algebras and highest weight categories.
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition