A128564 Triangle, read by rows, where T(n,k) equals the number of permutations of {1..n+1} with [(nk+k)/2] inversions for n>=k>=0.
1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 9, 22, 15, 1, 1, 29, 90, 90, 29, 1, 1, 49, 359, 573, 359, 98, 1, 1, 174, 1415, 3450, 3450, 1415, 174, 1, 1, 285, 5545, 17957, 29228, 21450, 5545, 628, 1, 1, 1068, 21670, 110010, 230131, 230131, 110010, 21670, 1068, 1
Offset: 0
Examples
Row sums equal 2*n! for n>0: [1, 2, 4, 12, 48, 240, 1440, 10080, 80640, ..., 2*n!,...]. Triangle begins: 1; 1, 1; 1, 2, 1; 1, 5, 5, 1; 1, 9, 22, 15, 1; 1, 29, 90, 90, 29, 1; 1, 49, 359, 573, 359, 98, 1; 1, 174, 1415, 3450, 3450, 1415, 174, 1; 1, 285, 5545, 17957, 29228, 21450, 5545, 628, 1; 1, 1068, 21670, 110010, 230131, 230131, 110010, 21670, 1068, 1; 1, 1717, 84591, 526724, 1729808, 2409581, 1729808, 686763, 84591, 4015, 1; ...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
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Maple
b:= proc(u, o) option remember; expand(`if`(u+o=0, 1, add(b(u+j-1, o-j)*x^(u+j-1), j=1..o)+ add(b(u-j, o+j-1)*x^(u-j), j=1..u))) end: T:= (n, k)-> coeff(b(n+1, 0), x, iquo((n+1)*k, 2)): seq(seq(T(n,k), k=0..n), n=0..10); # Alois P. Heinz, May 02 2017 -
Mathematica
b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1, Sum[b[u + j - 1, o - j]* x^(u+j-1), {j, 1, o}] + Sum[b[u-j, o+j-1]*x^(u-j), {j, 1, u}]]]; T[n_, k_] := Coefficient[b[n+1, 0], x, Quotient[(n+1)*k, 2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *) -
PARI
{T(n,k)=local(faq=prod(j=1, n+1, (1-q^j)/(1-q))); polcoeff(faq, (n*k+k)\2, q)}
Formula
T(n,k) = A008302(n+1, [(nk+k)/2]) = coefficient of q^[(nk+k)/2] in the q-factorial of n+1 for n>=0.
Comments