A350665 Number of permutations avoiding 321 of length 3n composed of only 3-cycles.

1, 2, 10, 60, 388, 2606, 17890, 124512, 874562, 6182198, 43903044, 312843918, 2235028210, 15999423988, 114710881886, 823463493632, 5917220509358

Offset: 0

Author

Kassie Archer, Jan 10 2022

Keywords

• nonn
• more

Comments

Sum over all Dyck paths D of L(D)*2^h(D), where h(D) is the number of times the Dyck path hits the x-axis and L(D) is the product of binomial coefficients (u_i(D)+d_i(D) choose u_i(D)), where u_i(D) is the number of up-steps between the i-th and (i+1)-st down step and d_i(D) is the number of down-steps between the i-th and (i+1)-st up step.

Examples

			For n=2, the ten permutations (in one-line notation and cycle notation) are:
  [2, 3, 1, 5, 6, 4] (1,2,3)(4,5,6)
  [3, 1, 2, 5, 6, 4] (1,3,2)(4,5,6)
  [2, 3, 1, 6, 4, 5] (1,2,3)(4,6,5)
  [3, 1, 2, 6, 4, 5] (1,3,2)(4,6,5)
  [4, 1, 6, 2, 3, 5] (1,4,2)(3,6,5)
  [2, 4, 6, 1, 3, 5] (1,2,4)(3,6,5)
  [4, 1, 5, 2, 6, 3] (1,4,2)(3,5,6)
  [5, 6, 1, 2, 3, 4] (1,5,3)(2,6,4)
  [2, 4, 5, 1, 6, 3] (1,2,4)(3,5,6)
  [3, 4, 5, 6, 1, 2] (1,3,5)(2,4,6)
		

Links

• Kassie Archer and Christina Graves, Pattern-restricted permutations composed of 3-cycles, arXiv:2104.12664 [math.CO], 2021.

Crossrefs

Cf. A350645.