A093789 Hook products of all partitions of 10.
4725, 6400, 6400, 6912, 6912, 8064, 8064, 8100, 10368, 10368, 11520, 11520, 12096, 12096, 12600, 12600, 14400, 14400, 16128, 16128, 17280, 17280, 22680, 22680, 28800, 28800, 40320, 40320, 43200, 43200, 48384, 48384, 86400, 86400, 100800, 100800, 103680, 103680, 403200, 403200, 3628800, 3628800
Offset: 1
Programs
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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(10)[q])),q=1..numbpart(10))]);
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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[10!/T[10]] (* Jean-François Alcover, Aug 12 2024, after Alois P. Heinz in A060240 *)
Formula
a(n) = 10!/A003874(43-n).
Comments