OFFSET
0,5
COMMENTS
w(p) is defined (by Edelman, Simion and White) in the following way: if p = (c[1])(c[2])... is expressed in standard cycle form (i.e., cycles ordered by increasing smallest elements with each cycle written with its smallest element in the first position), then w(p) = 0*|c[1]| + 1*|c[2]| + 2*|c[3]| + ..., where |c[j]| denotes the number of entries in the cycle c[j].
LINKS
Alois P. Heinz, Rows n = 0..50, flattened
P. H. Edelman, R. Simion and D. White, Partition statistics on permutations, Discrete Math. 99 (1992), 63-68.
M. Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.
FORMULA
Generating polynomial of row n is P[n](t) = Product_{i=0..n-1} (i + t^i).
Sum_{k=0..n*(n-1)/2} (k+1) * T(n,k) = A121586(n). - Alois P. Heinz, May 04 2023
EXAMPLE
T(4,2)=3 because we have w(1423) = w((1)(243)) = 0*1 + 1*3 = 3, w(1342) = w((1)(234)) = 0*1 + 1*3=3 and w(2134) = w((12)(3)(4)) = 0*2 + 1*1 + 2*1 = 3.
Triangle starts:
1;
1;
1, 1;
2, 2, 1, 1;
6, 6, 3, 5, 2, 1, 1;
24, 24, 12, 20, 14, 10, 7, 5, 2, 1, 1;
MAPLE
for n from 0 to 8 do P[n]:=sort(expand(product(i+t^i, i=0..n-1))) od: for n from 0 to 8 do seq(coeff(P[n], t, j), j=0..n*(n-1)/2) od; # yields sequence in triangular form
# second Maple program:
p:= proc(n) option remember; `if`(n<0, 1, expand((n+t^n)*p(n-1))) end:
T:= n-> (h-> seq(coeff(h, t, i), i=0..degree(h)))(p(n-1)):
seq(T(n), n=0..8); # Alois P. Heinz, Dec 16 2016
MATHEMATICA
p[n_] := p[n] = If[n<0, 1, Expand[(n+t^n)*p[n-1]]]; T[n_] := Function[h, Table[Coefficient[h, t, i], {i, 0, Exponent[h, t]}]][p[n-1]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Apr 11 2007
EXTENSIONS
One term for row n=0 prepended by Alois P. Heinz, Dec 16 2016
STATUS
approved