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A344898
Number of equivalence classes of pairs of permutations on S2n where 2 pairs are equivalent if they generate similar maps on Dyck paths.
0
1, 3, 154, 8369, 711226, 90349957, 16012077362, 3768789527617, 1136241039871954, 426747190631335301, 195301450278484563322, 106968871128338892427537, 69076413764424335543681642, 51931946172675368683512111589, 44964793280161619728525791864226, 44419470206051792513510236597094657
OFFSET
1,2
LINKS
Kevin Limanta, Hopein Christofen Tang, and Yozef Tjandra, Permutation-generated maps between Dyck paths, arXiv:2105.14439 [math.CO], 2021.
FORMULA
a(n) = 1 - n^2 + 2*Sum_{a=1, n-1} Sum_{b=1, n-1} n!^2*(binomial(2*n-2-a-b, n-2)+binomial(2*n-2-a-b, n-1-a))/(max(a,2)!*max(b,2)!) for n>=3. - clarified by Kevin Limanta, Dec 29 2022
a(n) ~ c * 2^(2*n + 1) * n^(2*n + 1/2) / exp(2*n), where c = sqrt(Pi) * (25/16 + exp(1) - 5*exp(1/2)/2) = 0.281782323432896188420860093697452373839427854773... - Vaclav Kotesovec, Dec 29 2022
MATHEMATICA
Join[{1, 3}, Table[1 - n^2 + 2*Sum[Sum[n!^2*(Binomial[2*n - 2 - k - j, n - 2] + Binomial[2*n - 2 - k - j, n - 1 - k])/(Max[k, 2]! * Max[j, 2]!), {k, 1, n - 1}], {j, 1, n - 1}], {n, 3, 20}]] (* Vaclav Kotesovec, Dec 29 2022 *)
PROG
(PARI) a(n) = if (n==1, 1, if (n==2, 3, 1 - n^2 + 2*sum(a=1, n-1, sum(b=1, n-1, n!^2*(binomial(2*n-2-a-b, n-2)+binomial(2*n-2-a-b, n-1-a))/(max(a, 2)!*max(b, 2)!))))); \\ corrected by Michel Marcus, Dec 29 2022
CROSSREFS
Sequence in context: A039934 A156990 A075514 * A327058 A087306 A278877
KEYWORD
nonn
AUTHOR
Michel Marcus, Jun 01 2021
EXTENSIONS
Offset 1 from Vaclav Kotesovec, Dec 29 2022
STATUS
approved