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It is a wqo as well. Posets and tournaments. U is the set of all pairs S = ([n], ≤S ) where ≤S is a non-strict partial ordering on [n]. We set R = ([m], ≤R ) ≺ S = ([n], ≤S ) if and only if there is an injection f : [m] → [n] such that x ≤R y ⇐⇒ f (x) ≤S f (y) for every x, y in [m]. Thus R ≺ S means that the poset R is an induced subposet of S. Downsets in (U, ≺ ), hereditary properties of posets, and their growths were investigated by Balogh, Bollob´ as and Morris in . For the unlabeled count they obtained the following result.
Preprint.  J. Balogh, B. Bollob´ a s, and M. Simonovits. The number of graphs without forbidden subgraphs. J. Combin. Theory Ser. B, 91(1):1–24, 2004.  J. Balogh, B. Bollob´ a s, and D. Weinreich. The speed of hereditary properties of graphs. J. Combin. Theory Ser. B, 79(2):131–156, 2000.  J. Balogh, B. Bollob´ a s, and D. Weinreich. The penultimate rate of growth for graph properties. European J. , 22(3):277–289, 2001.  J. Balogh, B. Bollob´ a s, and D. Weinreich. Measures on monotone properties of graphs.
22(3):277–289, 2001.  J. Balogh, B. Bollob´ a s, and D. Weinreich. Measures on monotone properties of graphs. Discrete Appl. , 116(1-2):17–36, 2002.  J. Balogh, B. Bollob´ a s, and D. Weinreich. A jump to the Bell number for hereditary graph properties. J. Combin. Theory Ser. B, 95(1):29–48, 2005.  E. Barcucci, A. Del Lungo, A. Frosini, and S. Rinaldi. From rational functions to regular languages. In Formal power series and algebraic combinatorics (Moscow, 2000), pages 633–644. Springer, Berlin, 2000.