A093787 Hook products of all partitions of 9.

1680, 1680, 1920, 1920, 2160, 2160, 2240, 2240, 3024, 3024, 3456, 3456, 4320, 4320, 5184, 6480, 6480, 7560, 7560, 8640, 8640, 8640, 12960, 12960, 13440, 13440, 45360, 45360, 362880, 362880

Offset

1,1

Links

Table of n, a(n) for n=1..30.

Formula

a(n) = 9!/A003873(31-n).

Maple

H:=proc(pa) local F, j, p, Q, i, col, a, A: F:=proc(x) local i, ct: ct:=0: for i from 1 to nops(x) do if x[i]>1 then ct:=ct+1 else fi od: ct; end: for j from 1 to nops(pa) do p[1][j]:=pa[j] od: Q[1]:=[seq(p[1][j], j=1..nops(pa))]: for i from 2 to pa[1] do for j from 1 to F(Q[i-1]) do p[i][j]:=Q[i-1][j]-1 od: Q[i]:=[seq(p[i][j], j=1..F(Q[i-1]))] od: for i from 1 to pa[1] do col[i]:=[seq(Q[i][j]+nops(Q[i])-j, j=1..nops(Q[i]))] od: a:=proc(i, j) if i<=nops(Q[j]) and j<=pa[1] then Q[j][i]+nops(Q[j])-i else 1 fi end: A:=matrix(nops(pa), pa[1], a): product(product(A[m, n], n=1..pa[1]), m=1..nops(pa)); end: with(combinat): rev:=proc(a) [seq(a[nops(a)+1-i], i=1..nops(a))] end: sort([seq(H(rev(partition(9)[q])), q=1..numbpart(9))]);

Mathematica

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i < 1, 0, Flatten@Table[g[n - i*j, i - 1, Join[l, Array[i&, j]]], {j, 0, n/i}]]];
T[n_] := g[n, n, {}];
Sort[9!/T[9]] (* Jean-François Alcover, Aug 12 2024, after Alois P. Heinz in A060240 *)

Crossrefs

Cf. A003873, A060240.

Row n=9 of A093784.

Sequence in context: A159625 A156425 A247853 * A258920 A268288 A175749

Adjacent sequences: A093784 A093785 A093786 * A093788 A093789 A093790

Keyword

fini,full,nonn

Author

Emeric Deutsch, May 17 2004

Status

approved