A349413 Number of smooth positroid varieties corresponding to derangements in S_n.

1, 0, 1, 2, 5, 14, 40, 118, 357, 1100

Offset: 0

Author

Jordan Weaver, Nov 16 2021

Keywords

• nonn
• more

Comments

a(n) is also the number of derangements in S_n whose chordal diagram contains no crossed alignments.

a(n) is also the number of derangements in S_n whose chordal diagram is a separable union of star graphs, where a star graph is the chordal diagram of a permutation in S_m of the form w(i) = i + t (mod m) for some t.

a(n) counts the complement of A349456 in the set of all derangements of S_n (A000166).

a(n) appears to be the number of n-edge ordered trees in which each nonleaf has at least two children and each leftmost child has a designated favorite sibling. For example, for n = 3, the underlying tree must be a root with 3 children and there are two choices for the favorite sibling, so a(3) = 2. The generating function for these trees, A(x) = 1 + x^2 + 2*x^3 + 5*x^4 + ..., is easily shown, using the "symbolic method" of Flajolet and Sedgewick, to satisfy A(x) = 1 + x^2*A(x)^2/(1 - x*A(x))^2. - David Callan, May 15 2022

Examples

			For n=4, the a(4)=5 derangements in one-line notation are 2143, 4321, 2341, 4123, and 3412.
		

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. A000166, A349456, A349458, A349457.

Formula

a(n) = A000166(n) - A349456(n).