The Number of 0-1-2 Increasing Trees as Two Different Evaluations of the Tutte Polynomial of a Complete Graph

@article{Merino2008TheNO,
title={The Number of 0-1-2 Increasing Trees as Two Different Evaluations of the Tutte Polynomial of a Complete Graph},
author={C. Merino},
journal={Electr. J. Comb.},
year={2008},
volume={15}
}

If Tn(x, y) is the Tutte polynomial of the complete graph Kn, we have the equality Tn+1(1, 0) = Tn(2, 0). This has an almost trivial proof with the right combinatorial interpretation of Tn(1, 0) and Tn(2, 0). We present an algebraic proof of a result with the same flavour as the latter: Tn+2(1,−1) = Tn(2,−1), where Tn(1,−1) has the combinatorial interpretation of being the number of 0–1–2 increasing trees on n vertices.