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%I #20 Jul 19 2022 01:13:17
%S 0,0,0,0,4,70,825,8526,85372,870756
%N Number of singular positroids in the Grassmannian variety Gr(k,n) for a fixed n and any 0 <= k <= n.
%C a(n) is also the number of decorated permutations whose chordal diagram contains a crossed alignment.
%C a(n) counts the complement of A349458 in the set of all positroid varieties/decorated permutations on n elements (A000522).
%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) = Sum_{i=0..n} (2^i)*binomial(n,i)*b(n), where b(n) is the sequence A349456.
%F a(n) = A000522(n) - A349458(n).
%e For n = 4, the a(4) = 4 singular positroid varieties correspond to the decorated permutations whose underlying permutations are 2413, 3421, 3142, and 4312 in one-line notation. Note that none of these permutations contain fixed points, hence no decorations are needed.
%Y Cf. A000522, A349413, A349456, A349458.
%K nonn,more
%O 0,5
%A _Jordan Weaver_, Nov 17 2021
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