OFFSET
0,6
COMMENTS
Define the "entropy" (or variance) of a permutation pi to be Sum_{i=1..n} (pi(i)-i)^2 = A006331(n) - 2*Sum_i i*pi(i), as in A126972.
This characteristic is obviously an even number, 2*v(pi).
Row n of the triangle shows the statistics (frequency distribution) of v for the n! = A000142(n) possible permutations of [n].
T(n,0)=1 arises the identity permutation where v=0.
T(n,1)=n-1 arises from the n-1 different ways of creating an entropy of 2 by swapping a pair of adjacent entries in the identity permutation.
The final 1 in each row arises from the permutation with maximal entropy, that is the permutation with integers reversed relative to the identity permutation.
Removing zeros in A135298 creates a sequence which is similar in the initial terms, because contributions to A135298(n) stem from permutations of some unique [j] if n is not too large, which establishes a 1-to-1 correspondence between the term A006331(n)-2*sum_i i*pi(i) mentioned above and the defining formula in A135298.
The rows of this triangle have a geometric interpretation. Let P_n be the n-dimensional permutohedron, the Voronoi cell of the lattice A_n* (Conway-Sloane, 1993, p. 474), which is a polytope with (n+1)! vertices. Start at any vertex, and count how many vertices there are at squared-distance v from the starting vertex: this is T(n+1,v). For example, in three dimensions the permutohedron is a truncated octahedron, the squared distances from a vertex to all the vertices are (when suitably scaled) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and the numbers of vertices at these distances are 1, 3, 1, 4, 2, 2, 2, 4, 1, 3, 1, which is row 4 of the array. See Chap. 21, Section 3.F, op. cit., for further details. - N. J. A. Sloane, Oct 13 2015
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993.
LINKS
Alois P. Heinz, Rows n = 0..20, flattened
FindStat - Combinatorial Statistic Finder, The rank of the permutation inside the alternating sign matrix lattice.
FORMULA
Sum_{k>=0} k * T(n,k) = A001754(n+1). - Alois P. Heinz, Mar 02 2024
EXAMPLE
Triangle T(n,v) starts in row n=0 and column v=0 as follows:
1;
1;
1, 1;
1, 2, 0, 2, 1;
1, 3, 1, 4, 2, 2, 2, 4, 1, 3, 1;
1, 4, 3, 6, 7, 6, 4, 10, 6, 10, 6, 10, 6, 10, 4, 6, 7, 6, 3, 4, 1;
...
MAPLE
with(combinat):
T:= n-> (p-> seq(coeff(p, x, j), j=ldegree(p)..degree(p)))
(add(x^add(i*l[i], i=1..n), l=permute(n))):
seq(T(n), n=0..7); # Alois P. Heinz, Aug 28 2014
# second Maple program:
b:= proc(s) option remember; (n-> `if`(n=0, 1, add(expand(
x^((n-j)^2/2)*b(s minus {j})), j=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
seq(T(n), n=0..7); # Alois P. Heinz, Mar 02 2024
MATHEMATICA
b[s_] := b[s] = With[{n = Length[s]}, If[n == 0, 1, Sum[Expand[x^((n-j)^2/2)*b[s~Complement~{j}]], {j, s}]]];
T[n_] := CoefficientList[b[Range[n]], x];
Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 22 2024, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch and R. J. Mathar, Oct 22 2010
EXTENSIONS
Row length term corrected by R. J. Mathar, Oct 23 2010
T(0,0)=1 prepended by Alois P. Heinz, Nov 23 2023
STATUS
approved