A349458 Number of smooth positroids in the Grassmannian variety Gr(k,n) for a fixed n and any 0 <= k <= n.

1, 2, 5, 16, 61, 256, 1132, 5174, 24229, 115654, 560741, 2754082, 13674212, 68522208, 346100952, 1760213254, 9006390373, 46329244034, 239455376071, 1242923653316, 6476376834789, 33863408028888, 177625109853808, 934404580376016

Offset: 0

Author

Jordan Weaver, Nov 17 2021

Keywords

• nonn

Comments

a(n) is also the number of decorated permutations whose chordal diagram is a separable union of star graphs.

a(n) is also the number of decorated permutations whose chordal diagram contains no crossed alignments.

a(n) counts the complement of A349457 in the set of all positroid varieties/decorated permutations on n elements (A000522).

Examples

			For n = 3, the a(3) = 16 positroids correspond the decorated permutations with underlying permutations 231, 312, 321, 213, 132, and 123 in one-line notation. Each fixed point, e.g., the 2 in 321, can be colored in two ways. Hence 321, 213, and 132 contribute 2 decorated permutations each, 123 contributes 8, while 231 and 312 each contribute 1.
		

Links

• Jordan Weaver, Table of n, a(n) for n = 0..50
• 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, A349457.

Formula

a(n) = Sum_{i=0..n} (2^i)*binomial(n,i)*b(n), where b(n) is the sequence A349413.

a(n) = A000522(n) - A349457(n).

Extensions

a(10)-a(23) from Jordan Weaver, Apr 19 2022