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

0,2

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).

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.

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).

Example

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.

Crossrefs

Cf. A000522, A349413, A349456, A349457.

Sequence in context: A012051 A012159 A009736 * A370797 A307228 A104858

Adjacent sequences: A349455 A349456 A349457 * A349459 A349460 A349461

Keyword

nonn

Author

Jordan Weaver, Nov 17 2021

Extensions

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

Status

approved