A334623
Sum of the n-th powers of the descent set statistics for permutations of [n].
Original entry on oeis.org
1, 1, 2, 18, 1576, 2675100, 128235838496, 265039489112493900, 31306198216486969509375104, 278983981168082455883720325976751040, 235157286166918393786165504356030195355598048512, 23075317400822150539572583950910707053701314350537805923757600
Offset: 0
-
b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
add(b(u-j, o+j-1, t+1)*x^floor(2^(t-1)), j=1..u)+
add(b(u+j-1, o-j, t+1), j=1..o)))
end:
a:= n-> (p-> add(coeff(p, x, i)^n, i=0..degree(p)))(b(n, 0$2)):
seq(a(n), n=0..12);
-
b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1,
Sum[b[u - j, o + j - 1, t + 1]*x^Floor[2^(t - 1)], {j, 1, u}] +
Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]]];
a[n_] := Function[p, Sum[Coefficient[p, x, i]^n, {i, 0, Exponent[p, x]}]][ b[n, 0, 0]];
a /@ Range[0, 12] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)
A335545
A(n,k) is the sum of the k-th powers of the (positive) number of permutations of [n] with j descents (j=0..max(0,n-1)); square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 6, 4, 1, 1, 2, 18, 24, 5, 1, 1, 2, 66, 244, 120, 6, 1, 1, 2, 258, 2664, 5710, 720, 7, 1, 1, 2, 1026, 29284, 322650, 188908, 5040, 8, 1, 1, 2, 4098, 322104, 19888690, 55457604, 8702820, 40320, 9, 1, 1, 2, 16386, 3543124, 1276095330, 16657451236, 17484605040, 524888040, 362880, 10
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, ...
3, 6, 18, 66, 258, 1026, ...
4, 24, 244, 2664, 29284, 322104, ...
5, 120, 5710, 322650, 19888690, 1276095330, ...
6, 720, 188908, 55457604, 16657451236, 5025377832180, ...
...
-
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
expand(add(b(u-j, o+j-1, 1)*x^t, j=1..u))+
add(b(u+j-1, o-j, 1), j=1..o))
end:
A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
A:= (n, k)-> add(combinat[eulerian1](n, j)^k, j=0..max(0, n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
-
B[n_, k_] := B[n, k] = Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
A[0, ] = 1; A[n, k_] := Sum[B[n, j]^k, {j, 0, n-1}];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)
Showing 1-2 of 2 results.