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A225725 Triangle of transformation semigroup sizes generated by a single element.

%I A225725 #41 Jan 06 2025 22:27:03
%S A225725 1,1,3,1,10,15,2,41,129,80,6,196,1115,1260,510,24,20,1057,10395,17780,
%T A225725 12840,3744,840,6322,105315,258510,264810,135492,47250,4920,0,0,504,0,
%U A225725 420,41393,1160635,4018000,5318180,3788400,1837024,513120,38640,0,32256,0,26880,0,0,2688
%N A225725 Triangle of transformation semigroup sizes generated by a single element.
%C A225725 If you take the powers of a finite function you generate a lollipop graph. A222029 organizes the lollipops by cycle size. The table organized by total lollipop size with the tail included is this triangle.
%H A225725 Chad Brewbaker, <a href="/A225725/a225725.txt">Ruby program for A225725</a>
%e A225725 T(1,1) = #{[0]} = 1.
%e A225725 T(2,1) = #{[0,1], [0,0], [1,1]} = 3.
%e A225725 T(2,2) = #{[1,0]} = 1.
%e A225725 Triangle begins:
%e A225725 :    1;
%e A225725 :    1;
%e A225725 :    3,      1;
%e A225725 :   10,     15,      2;
%e A225725 :   41,    129,     80,      6;
%e A225725 :  196,   1115,   1260,    510,     24,    20;
%e A225725 : 1057,  10395,  17780,  12840,   3744,   840;
%e A225725 : 6322, 105315, 258510, 264810, 135492, 47250, 4920, 0, 0, 504, 0, 420;
%o A225725 (Ruby) # See Brewbaker link.
%Y A225725 First column is A000248.
%Y A225725 Row sums are: A000312.
%Y A225725 Row lengths are A000793.
%Y A225725 Number of nonzero elements of rows give A009490.
%Y A225725 Cf. A222029.
%K A225725 nonn,tabf
%O A225725 0,3
%A A225725 _Chad Brewbaker_, May 14 2013
%E A225725 More terms, some terms corrected by _Alois P. Heinz_, Aug 17 2017