cp's OEIS Frontend

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

%I A349458 #28 Jul 19 2022 01:13:13
%S A349458 1,2,5,16,61,256,1132,5174,24229,115654,560741,2754082,13674212,
%T A349458 68522208,346100952,1760213254,9006390373,46329244034,239455376071,
%U A349458 1242923653316,6476376834789,33863408028888,177625109853808,934404580376016
%N A349458 Number of smooth positroids in the Grassmannian variety Gr(k,n) for a fixed n and any 0 <= k <= n.
%C A349458 a(n) is also the number of decorated permutations whose chordal diagram is a separable union of star graphs.
%C A349458 a(n) is also the number of decorated permutations whose chordal diagram contains no crossed alignments.
%C A349458 a(n) counts the complement of A349457 in the set of all positroid varieties/decorated permutations on n elements (A000522).
%H A349458 Jordan Weaver, <a href="/A349458/b349458.txt">Table of n, a(n) for n = 0..50</a>
%H A349458 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 A349458 S. Corteel, <a href="https://arxiv.org/abs/math/0601469">Crossings and alignments of permutations</a>, arXiv:math/0601469 [math.CO], 2006.
%H A349458 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 A349458 A. Postnikov, <a href="https://arxiv.org/abs/math/0609764">Total positivity, Grassmannians, and networks</a>, arXiv:math/0609764 [math.CO], 2006.
%F A349458 a(n) = Sum_{i=0..n} (2^i)*binomial(n,i)*b(n), where b(n) is the sequence A349413.
%F A349458 a(n) = A000522(n) - A349457(n).
%e A349458 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.
%Y A349458 Cf. A000522, A349413, A349456, A349457.
%K A349458 nonn
%O A349458 0,2
%A A349458 _Jordan Weaver_, Nov 17 2021
%E A349458 a(10)-a(23) from _Jordan Weaver_, Apr 19 2022