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A275550
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Number of classes of endofunctions of [n] under reversal and complement to n+1.
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14
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1, 1, 2, 10, 72, 819, 11772, 206572, 4196352, 96871525, 2500050000, 71328400806, 2229026605056, 75718793541895, 2778001759096256, 109473473278652344, 4611686020574871552, 206810065502975099529
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OFFSET
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0,3
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COMMENTS
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Possible classes size are 1,2,4
n 1 2 4
-----------------
1 1 0 0
2 0 2 0
3 1 5 4
4 0 16 56
5 1 74 744
6 0 216 11556
7 1 1371 205200.
Classes of size 2 can be further decomposed by whether the function is stable by reversal or stable by (reversal and complement).
n 2 2-r 2-rc
-----------------
1 0 0 0
2 2 1 1
3 5 4 1
4 16 8 8
5 74 62 12
6 216 108 108
7 1371 1200 171.
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LINKS
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FORMULA
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a(n) = (1+(-1)^(n+1)+2*n^n+(3+((-1)^(n+1))*(n-1)+n)*n^(floor(n/2)) )/8.
Classes of size 2: (2 (-1 + (-1)^n) + n^floor(n/2)*(3 + ((-1)^(1 + n))* (-1 + n) + n))/4.
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MATHEMATICA
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Table[1/8 (1+(-1)^(1+n)+2 n^n+n^Floor[n/2] (3+(-1)^(n+1) (-1+n)+n)), {n, 1, 17}]
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PROG
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(PARI) a(n) = (1+(-1)^(n+1)+2*n^n+(3+((-1)^(n+1))*(n-1)+n)*n^(floor(n/2)) )/8; \\ Andrew Howroyd, Sep 30 2017
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CROSSREFS
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Cf. A000169 Classes under translation mod n
Cf. A168658 Classes under complement to n+1
Cf. A130293 Classes under translation and rotation
Cf. A081721 Classes under rotation and reversal
Cf. A275550 Classes under reversal and complement
Cf. A275551 Classes under translation and reversal
Cf. A275552 Classes under translation and complement
Cf. A275553 Classes under translation, complement and reversal
Cf. A275554 Classes under translation, rotation and complement
Cf. A275555 Classes under translation, rotation and reversal
Cf. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal
Cf. A192396 floor(((k+1)^n-(1+(-1)^k)/2)/2)
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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