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Number of smooth positroid varieties corresponding to derangements in S_n.
5

%I #41 Jul 19 2022 01:13:27

%S 1,0,1,2,5,14,40,118,357,1100

%N Number of smooth positroid varieties corresponding to derangements in S_n.

%C a(n) is also the number of derangements in S_n whose chordal diagram contains no crossed alignments.

%C a(n) is also the number of derangements in S_n whose chordal diagram is a separable union of star graphs, where a star graph is the chordal diagram of a permutation in S_m of the form w(i) = i + t (mod m) for some t.

%C a(n) counts the complement of A349456 in the set of all derangements of S_n (A000166).

%C a(n) appears to be the number of n-edge ordered trees in which each nonleaf has at least two children and each leftmost child has a designated favorite sibling. For example, for n = 3, the underlying tree must be a root with 3 children and there are two choices for the favorite sibling, so a(3) = 2. The generating function for these trees, A(x) = 1 + x^2 + 2*x^3 + 5*x^4 + ..., is easily shown, using the "symbolic method" of Flajolet and Sedgewick, to satisfy A(x) = 1 + x^2*A(x)^2/(1 - x*A(x))^2. - _David Callan_, May 15 2022

%H Sara C. Billey and Jordan E. Weaver, <a href="https://arxiv.org/abs/2207.06508">Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs</a>, arXiv:2207.06508 [math.CO], 2022.

%H S. Corteel, <a href="https://arxiv.org/abs/math/0601469">Crossings and alignments of permutations</a>, arXiv:math/0601469 [math.CO], 2006.

%H A. Knutson, T. Lam and D. Speyer, <a href="http://dx.doi.org/10.1112/S0010437X13007240">Positroid varieties: juggling and geometry</a>, Compos. Math. 149 (2013), no. 10, 1710-1752.

%H A. Postnikov, <a href="https://arxiv.org/abs/math/0609764">Total positivity, Grassmannians, and networks</a>, arXiv:math/0609764 [math.CO], 2006.

%F a(n) = A000166(n) - A349456(n).

%e For n=4, the a(4)=5 derangements in one-line notation are 2143, 4321, 2341, 4123, and 3412.

%Y Cf. A000166, A349456, A349458, A349457.

%K nonn,more

%O 0,4

%A _Jordan Weaver_, Nov 16 2021