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A093787 Hook products of all partitions of 9.

%I A093787 #11 Sep 23 2024 14:10:41
%S A093787 1680,1680,1920,1920,2160,2160,2240,2240,3024,3024,3456,3456,4320,
%T A093787 4320,5184,6480,6480,7560,7560,8640,8640,8640,12960,12960,13440,13440,
%U A093787 45360,45360,362880,362880
%N A093787 Hook products of all partitions of 9.
%F A093787 a(n) = 9!/A003873(31-n).
%p A093787 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))]);
%t A093787 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 A093787 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 A093787 T[n_] := g[n, n, {}];
%t A093787 Sort[9!/T[9]] (* _Jean-François Alcover_, Aug 12 2024, after _Alois P. Heinz_ in A060240 *)
%Y A093787 Cf. A003873, A060240.
%Y A093787 Row n=9 of A093784.
%K A093787 fini,full,nonn
%O A093787 1,1
%A A093787 _Emeric Deutsch_, May 17 2004