A348634 Number of transitive relations on an n-set with exactly five ordered pairs.

0, 0, 0, 27, 768, 8771, 63468, 340620, 1470784, 5371002, 17153352, 49075521, 128066400, 309124101, 697874996, 1486830618, 3011414784, 5833686340, 10863883728, 19532496375, 34028554944, 57623258007, 95101946940, 153331834040, 241997811264, 374544148830, 569365964440, 851301035325, 1253479866912, 1819599953913, 2606698902276

Offset

0,4

Links

Table of n, a(n) for n=0..30.

Formula

a(n) = 27*C(n,3) + 660*C(n,4) + 5201*C(n,5) + 21822*C(n,6) + 54600*C(n,7) + 84000*C(n,8) + 75600*C(n,9) + 30240*C(n,10).

a(n) = (1/120)*(n^10 - 20*n^9 + 220*n^8 - 1500*n^7 + 6710*n^6 - 19954*n^5 + 38765*n^4 - 46950*n^3 + 31944*n^2 - 9216*n).

a(n) = C(n,3)*(n^7 - 17*n^6 + 167*n^5 - 965*n^4 + 3481*n^3 - 7581*n^2 + 9060*n - 4608)/20. - Chai Wah Wu, Jan 06 2022

Example

No relation containing exactly five ordered pairs on a 2-element set exists. Thus a(2)=0.
Also, there are 27 transitive relations with exactly five ordered pairs on a 3-set. One such relation is {(1,1),(1,2),(1,3),(2,2),(3,2)} on the 3-set {1,2,3}.

Prog

(Python)
def A348634(n): return n*(n - 2)*(n - 1)*(n*(n*(n*(n*(n*(n*(n - 17) + 167) - 965) + 3481) - 7581) + 9060) - 4608)//120 # Chai Wah Wu, Jan 06 2022

Crossrefs

Cf. A349919, A349927, A349849, A006905.

Sequence in context: A284039 A159668 A158645 * A232951 A138979 A183505

Adjacent sequences: A348631 A348632 A348633 * A348635 A348636 A348637

Keyword

nonn

Author

Firdous Ahmad Mala, Dec 13 2021

Extensions

a(9) corrected by Georg Fischer, Mar 19 2023

Status

approved