# [seqfan] number of k-cycles for n-simplex, n-cube, n-orthoplex (Also Errors in A085452)

Andrew Weimholt andrew.weimholt at gmail.com
Sun Nov 15 07:58:58 CET 2009

```Number of k-cycles in complete graph on n nodes (graph of (n-1)-simplex)

Triangle starts at row n=3 and col k=3

row 3: ... 1
row 4: ... 4, 3
row 5: ... 10, 15, 12
row 6: ... 20, 45, 72, 60
row 7: ... 35, 105, 252, 420, 360
row 8: ... 56, 210, 672, 1680, 2880, 2520

Looks like this is contained in the triangle in A144151, which
includes additional rows and columns (n>=0,k>=0)

Number of k-cycles on the graph of an n-orthoplex

Triangle Starts at n=2, k=3  (2n-2 elements in row n)

row 2: ... 0, 1
row 3: ... 8, 15, 24, 16
row 4: ... 32, 102, 288, 640, 960, 744
row 5: ... 80, 370, 1584, 5920, 18240, 43080, 69120, 56256

Not in OEIS

Number of k-cycles on the graph of an n-cube

Triangle starts at n=2, k=3  (2^n - 2 elements in row n)

row 2: ... 0, 1
row 3: ... 0, 6, 0, 16, 0, 6
row 4: ... 0, 24, 0, 128, 0, 696, 0, 2112, 0, 5024, 0, 5736, 0, 1344
row 5: ... 0, 80, 0, 640, 0, 6720, 0, 68736, 0, 591200, 0, 4652160, 0,
32146800, 0, ...(row not finished).

Alternatively:
Number of 2k-cycles on graph of an n-cube:

Triangle starts at n=2, k=3  (2^(n-1) - 1 elements in row n)

row 2: ... 1
row 3: ... 6, 16, 6
row 4: ... 24, 128, 696, 2112, 5024, 5736, 1344
row 5: ... 80, 640, 6720, 68736, 591200, 4652160, 32146800, ...(row
not finished).

Looks like this is A085452, which has errors

A085452 contains a "5" in between the end of row 4 and the beginning of row 5
(he probably accidentally copied the row number when he submitted the
sequence).

Also the title is slightly wrong:

"Triangle T(n,k) read by rows: T(n,k) = number of cycles of length 2k
in the binary n-cube, for n >= 2, k = 2, 4, ..., 2^n."

says length "2k", but then says "k = 2, 4, ..."

Should keep the "2k" and change second part to "k=2, 3, ..., 2^(n-1)"

Andrew

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