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A082914 a(n) = n! / A003040(n).

%I A082914 #17 Feb 16 2025 08:32:49
%S A082914 1,2,3,8,20,45,144,448,1680,4725,17280,62208,290304,1254400,4465125,
%T A082914 18144000,72990720,391910400,1881169920,9754214400,45660160000,
%U A082914 205752960000,905748480000,5280992640000,28326238617600,162956344320000,853298675712000,5309413982208000
%N A082914 a(n) = n! / A003040(n).
%C A082914 Minimum product of hook lengths of a partition of n. - _Eric M. Schmidt_, May 07 2013
%H A082914 Eric M. Schmidt, <a href="/A082914/b082914.txt">Table of n, a(n) for n = 1..80</a>
%H A082914 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HookLengthFormula.html">Hook Length Formula</a>
%e A082914 For n=4, we can have
%e A082914 abcd, abc and ab (the rest are symmetric).
%e A082914 ......d.......cd
%e A082914 The hook products are 4! = 24, 4*2*1*1 = 8 and 3*2*2*1 = 12, so a(4) = 8. - _Jon Perry_
%p A082914 # Maple code from  _Emeric Deutsch_, May 12 2004 (Start)
%p A082914 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:
%p A082914 for j from 1 to nops(pa) do p[1][j]:=pa[j] od: Q[1]:=[seq(p[1][j],j=1..nops(pa))]:
%p A082914 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:
%p A082914 Q[i]:=[seq(p[i][j],j=1..F(Q[i-1]))] od:
%p A082914 for i from 1 to pa[1] do col[i]:=[seq(Q[i][j]+ nops(Q[i])-j,j=1..nops(Q[i]))] od:
%p A082914 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:
%p A082914 A:=matrix(nops(pa),pa[1],a): product(product(A[m,n],n=1..pa[1]),m=1..nops(pa)); end:
%p A082914 with(combinat):
%p A082914 rev:=proc(a) [seq(a[nops(a)+1-i],i=1..nops(a))] end:
%p A082914 seq(sort([seq(H(rev(partition(j)[i])),i=1..numbpart(j))])[1],j=1..30);
%p A082914 # the procedure H gives the hook product for a given partition written with parts in nonincreasing order;
%p A082914 # if in the definition of the procedure a we replace "else 1" by "else x", then the matrix A yields all the hooklengths corresponding to a partition.
%p A082914 # (End)
%Y A082914 Cf. A003040.
%K A082914 nonn
%O A082914 1,2
%A A082914 Yuval Dekel (dekelyuval(AT)hotmail.com), May 25 2003
%E A082914 More terms from _Eric M. Schmidt_, May 07 2013