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A348634
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Number of transitive relations on an n-set with exactly five ordered pairs.
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1
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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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
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EXAMPLE
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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}.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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