On Dedekind Numbers and Two Sequences of Knuth
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
Chicago, IL 60607
We consider the sequence whose nth term is the number F(n) of anti-chains
in the partially ordered set whose elements are 0, 1, ..., n–1
and the order relation is coordinate-wise on the binary representation
of each integer. This sequence is a sort of "background" sequence to
its more prominent subsequence of Dedekind numbers, that is, the sequence
whose terms are F(2k).
We also consider the sequence of first differences
with terms F(n) – F(n–1).
We discuss, state, and prove some
(recursive) relations between the terms of these three sequences.
Full version: pdf,
(Concerned with sequences
Received July 13 2021; revised versions received December 23 2021; December 27 2021.
Published in Journal of Integer Sequences,
December 27 2021.
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