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Triangle read by rows: T(n,k) is the number of even permutations (of an n-set) with exactly k fixed points.
3

%I #22 Aug 29 2024 02:11:27

%S 1,0,1,0,0,1,2,0,0,1,3,8,0,0,1,24,15,20,0,0,1,130,144,45,40,0,0,1,930,

%T 910,504,105,70,0,0,1,7413,7440,3640,1344,210,112,0,0,1,66752,66717,

%U 33480,10920,3024,378,168,0,0,1

%N Triangle read by rows: T(n,k) is the number of even permutations (of an n-set) with exactly k fixed points.

%H Bashir Ali and A. Umar, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_200805&amp;filename=Some Combinatorial Properties of the Alternating Group.pdf">Some combinatorial properties of the alternating group</a>, Southeast Asian Bulletin Math. 32 (2008), 823-830.

%F T(n,k) = C(n,k)*A003221(n-k).

%F E.g.f.: (x^k(1-x^2/2) e^(-x))/k!(1-x).

%F T(n,k) + A145225(n,k) = A008290(n,k). - _R. J. Mathar_, Jul 06 2023

%F T(n,k) = (A008290(n,k) + A055137(n,k))/2. - _Julian Hatfield Iacoponi_, Aug 08 2024

%e Triangle starts:

%e 1;

%e 0, 1;

%e 0, 0, 1;

%e 2, 0, 0, 1;

%e 3, 8, 0, 0, 1;

%e 24, 15, 20, 0, 0, 1;

%e ...

%Y Row sums give A001710.

%Y Columns k=0..2 are A003221, A145219, A145220.

%Y Cf. A008290, A055137.

%K nonn,tabl

%O 0,7

%A _Abdullahi Umar_, Oct 09 2008