

A294789


Triangle read by rows: T(n,k), n>=2, 1 <= k <= n1, is the number of permutations in S_n in which there are k different values for the values mod n of the differences between adjacent elements when written in row notation.


0



2, 6, 0, 8, 0, 16, 20, 0, 100, 0, 12, 60, 288, 288, 72, 42, 0, 1764, 882, 2352, 0, 32, 96, 3584, 10112, 18816, 6912, 768, 54, 162, 12744, 39366, 156978, 105948, 47628, 0, 40, 760, 18000, 188400, 826400, 1420400, 966000, 194400, 14400
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OFFSET

2,1


COMMENTS

Take a permutation perm on the numbers 1 through n, think of it as a sequence: perm = (x1, x2, ... xn) where each of the x's is a number between 1 and n.
Now take the sequence of differences, read cyclically: Diff(perm) = (x2  x1, x3  x2, ... xn  x(n1), x1  xn) but take the differences mod n, so that we have no negative numbers, only numbers between 1 and n1.
Now consider Diff(perm) as a set, ignoring repetitions, and count how many different elements there are in it. Let that be called D(perm).
Among the n! different permutations on n elements, T(n,k) is the number with D(perm) = k.
Needs a bfile.


LINKS

Table of n, a(n) for n=2..46.
Vsevolod F. Lev, Sums and Differences Along Hamiltonian Cycles, arXiv:math/0601633 [math.CO], 2006.


EXAMPLE

For n=2 there are two permutations: {1,2} and {2,1} in each case there is but 1 difference, namely 1. This gives the first value of the sequence T(2,1)=2.
For n=3 there are six permutations and once again the only difference between successive member of the permutation is one. There are no successive members whose difference is two. This gives T(3,1)=6, T(3,2)=0.
The triangle begins:
2,
6, 0,
8, 0, 16,
20, 0, 100, 0,
12, 60, 288, 288, 72,
...
The row sums are n!.
The first column appears to be A002618.


MATHEMATICA

<< Combinatorica`;
For[n = 3, n <= 12, n++,
perm = Range[n];
For[i = 1, i <= n  1, i++, d[i] = 0];
set = {};
Print[]; Print[n];
For[index = 1, index <= n!, index++,
perm = NextPermutation[perm];
(*Print[perm[[index]]]; *)
set = {};
For[i = 1, i <= n  1, i++, diff = perm[[i + 1]]  perm[[i]];
If[diff < 0, diff = diff + n];
set = Union[set, {diff}]];
diff = perm[[1]]  perm[[n]];
If[diff < 0, diff = diff + n];
set = Union[set, {diff}];
L = Length[set];
d[L]++];
Print[Table[d[i], {i, 1, n  1}]]]


CROSSREFS

Cf. A000142, A002618.
Sequence in context: A241810 A156991 A229586 * A197035 A227805 A267314
Adjacent sequences: A294786 A294787 A294788 * A294790 A294791 A294792


KEYWORD

nonn,tabl


AUTHOR

David S. Newman, Nov 08 2017


EXTENSIONS

Edited by N. J. A. Sloane, Nov 11 2017


STATUS

approved



