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A093789 Hook products of all partitions of 10.

%I A093789 #11 Sep 23 2024 14:11:50
%S A093789 4725,6400,6400,6912,6912,8064,8064,8100,10368,10368,11520,11520,
%T A093789 12096,12096,12600,12600,14400,14400,16128,16128,17280,17280,22680,
%U A093789 22680,28800,28800,40320,40320,43200,43200,48384,48384,86400,86400,100800,100800,103680,103680,403200,403200,3628800,3628800
%N A093789 Hook products of all partitions of 10.
%C A093789 All 42 terms of this finite sequence are shown.
%F A093789 a(n) = 10!/A003874(43-n).
%p A093789 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))]);
%t A093789 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}]];
%t A093789 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 A093789 T[n_] := g[n, n, {}];
%t A093789 Sort[10!/T[10]] (* _Jean-François Alcover_, Aug 12 2024, after _Alois P. Heinz_ in A060240 *)
%Y A093789 Cf. A003874, A060240.
%Y A093789 Row n=10 of A093784.
%K A093789 fini,full,nonn
%O A093789 1,1
%A A093789 _Emeric Deutsch_, May 17 2004