A349457 Number of singular positroids in the Grassmannian variety Gr(k,n) for a fixed n and any 0 <= k <= n.

0, 0, 0, 0, 4, 70, 825, 8526, 85372, 870756

Offset: 0

Author

Jordan Weaver, Nov 17 2021

Keywords

• nonn
• more

Comments

a(n) is also the number of decorated permutations whose chordal diagram contains a crossed alignment.

a(n) counts the complement of A349458 in the set of all positroid varieties/decorated permutations on n elements (A000522).

Examples

			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.
		

Links

• Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.
• S. Corteel, Crossings and alignments of permutations, arXiv:math/0601469 [math.CO], 2006.
• A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.
• A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Crossrefs

Cf. A000522, A349413, A349456, A349458.

Formula

a(n) = Sum_{i=0..n} (2^i)*binomial(n,i)*b(n), where b(n) is the sequence A349456.

a(n) = A000522(n) - A349458(n).