A295929 - OEIS

%I #44 May 29 2018 20:37:33

%S 1,2,10,76,772,9856,152099,2758931,57602672,1362342830,36046013013,

%T 1056342305565,34002625115587,1193660155852584,45414253886783716,

%U 1862232981974586960,81893921416048297995,3845201559359081046971,192006280895048080286802

%N a(n) is the number of topological equivalence classes of excellent Morse functions on S^2 with 2n+2 critical points (n saddle points).

%C a(n) is also the number of ways of returning to an empty table for the first time after exactly 2n + 2 steps in the game of plates and olives. See the Carroll & Galvin link for a description of the game of plates and olives.

%D L. Nicolaescu, Counting Morse functions on the 2-sphere, Compositio Math. 144.

%H Andrew Howroyd, <a href="/A295929/b295929.txt">Table of n, a(n) for n = 0..50</a>

%H Teena Carroll, David Galvin, <a href="https://arxiv.org/abs/1711.10670">The game of plates and olives</a>, arXiv:1711.10670 [math.CO], 2017.

%H Andrew Howroyd, <a href="/A295929/a295929.txt">PARI Code</a>

%H L. Nicolaescu, <a href="https://www3.nd.edu/~lnicolae/MorseCountCompv3.pdf">Counting Morse functions on the 2-sphere</a>, Compositio Math. 144 (2008).

%F a(n) >= A001147(n) = (2*n - 1)!!. - _David A. Corneth_, Nov 30 2017

%e From _David A. Corneth_, Nov 30 2017: (Start)

%e a(0) = 1 as there is exactly one way to get an empty table for the first time in two steps:

%e Step 1: an empty plate is placed on the table.

%e Step 2: an empty plate is removed from the table. (End)

%Y Cf. A001147.

%K nonn

%O 0,2

%A _Michel Marcus_, Nov 30 2017

%E a(6)-a(18) from _Kyle Weingartner_, Dec 04 2017

%E New name from _Kyle Weingartner_, Dec 05 2017