A093791 Hook products of all partitions of 12.

62208, 82944, 82944, 85050, 85050, 107520, 107520, 115200, 115200, 129600, 129600, 134400, 134400, 136080, 136080, 155520, 155520, 161280, 161280, 179200, 179200, 181440, 201600, 201600, 226800, 226800, 228096, 230400, 230400, 248832

Offset: 1

Author

Emeric Deutsch, May 17 2004

Keywords

• fini
• full
• nonn

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..77

Crossrefs

Cf. A003876, A060240.

Row n=12 of A093784.

Programs

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(12)[q])),q=1..numbpart(12))]);

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[12!/T[12]] (* Jean-François Alcover, Sep 22 2024, after Alois P. Heinz in A060240 *)

Formula

a(n) = 12!/A003876(78-n).