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Number of singular positroid varieties corresponding to derangements in S_n.
3

%I #23 Jul 19 2022 01:13:21

%S 0,0,0,0,4,30,225,1736,14476,132396

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

%C a(n) is also the number of derangements whose chordal diagrams have crossed alignments.

%C a(n) counts the complement of A349413 in the set of all derangements of S_n (A000166).

%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) - A349413(n).

%e For n=4 the a(4)=4 derangements in one-line notation corresponding to singular positroid varieties are 2413, 3421, 3142, and 4312.

%Y Cf. A000166, A349413, A349457, A349458.

%K nonn,more

%O 0,5

%A _Jordan Weaver_, Nov 16 2021