On Solutions to a General Combinatorial Recurrence

  title={On Solutions to a General Combinatorial Recurrence},
  author={Michael Z. Spivey},
We develop techniques that can be applied to find solutions to the recurrence ∣∣n k ∣∣ = ( n+ k + )∣∣n−1 k ∣∣+ ( ′n+ ′k + ′)∣∣n−1 k−1∣∣+ [n = k = 0]. Many interesting combinatorial numbers, such as binomial coefficients, both kinds of Stirling and associated Stirling numbers, Lah numbers, Eulerian numbers, and second-order Eulerian numbers, satisfy special cases of this recurrence. Our techniques yield explicit expressions in the instances = − , = ′ = 0, and = ′ + 1, adding to the result of… CONTINUE READING

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