A generating function for triangular partitions
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- by L. Carlitz and Richard Scoville PDF
- Math. Comp. 29 (1975), 67-77 Request permission
Abstract:
Let ${T_k}(n)$ denote the number of solutions in nonnegative integers ${a_i}$, of the equation \[ n = \sum \limits _{i = 1}^k {\sum \limits _{j = 1}^{k - i + 1} {{a_{ij}}} } \] where the ${a_{ij}}$ satisfy the inequalities ${a_{ij}} \geqslant {a_{i + 1,j}},{a_{ij}} \geqslant {a_{i + 1,j - 1}}$. We show that \[ \sum \limits _{n = 1}^\infty {{T_k}(n){x^n} = {{(1 - x)}^{ - k}}{{(1 - {x^3})}^{ - k + 1}}{{(1 - {x^5})}^{ - k + 2}} \cdots {{(1 - {x^{2k - 1}})}^{ - 1}}.} \]References
- Ronald Alter, Some remarks and results on Catalan numbers, Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1971) Louisiana State Univ., Baton Rouge, La., 1971, pp. 109–132. MR 0329910
- L. Carlitz, Rectangular arrays and plane partitions, Acta Arith. 13 (1967/68), 29–47. MR 219430, DOI 10.4064/aa-13-1-29-47
- L. Carlitz, Sequences, paths, ballot numbers, Fibonacci Quart. 10 (1972), no. 5, 531–549. MR 317949 P. A. M. MACMAHON, Combinatory Analysis. Vol. 2, Cambridge, 1916.
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Math. Comp. 29 (1975), 67-77
- MSC: Primary 10A45
- DOI: https://doi.org/10.1090/S0025-5718-1975-0366803-0
- MathSciNet review: 0366803