A349849 -id:A349849 - cp's OEIS frontend

A349927 Number of transitive relations on an n-set with exactly three ordered pairs.

0, 0, 2, 43, 276, 1150, 3710, 10017, 23688, 50556, 99450, 183095, 319132, 531258, 850486, 1316525, 1979280, 2900472, 4155378, 5834691, 8046500, 10918390, 14599662, 19263673, 25110296, 32368500, 41299050, 52197327, 65396268, 81269426, 100234150

Offset: 0

Firdous Ahmad Mala, Dec 05 2021

			a(2) = 2. These two transitive relations are {(1,1),(1,2),(2,2)} and {(1,1),(2,1),(2,2)} on the 2-set {1,2}.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A349919, A349849.

A349927(n) = (1/6)*(n^6 - 6*n^5 + 24*n^4 - 47*n^3 + 38*n^2 - 10*n); \\ Antti Karttunen, Dec 05 2021

a(n) = 2*C(n,2) + 37*C(n,3) + 116*C(n,4) + 180*C(n,5) + 120*C(n,6).

a(n) = (1/6)*(n^6 - 6*n^5 + 24*n^4 - 47*n^3 + 38*n^2 - 10*n).

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

Original entry on oeis.org

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

Firdous Ahmad Mala, Dec 13 2021

nonn

			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}.

Cf. A349919, A349927, A349849, A006905.

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

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

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

cp's OEIS Frontend

A349927 Number of transitive relations on an n-set with exactly three ordered pairs.

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