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

1, 2, 3, 8, 20, 45, 144, 448, 1680, 4725, 17280, 62208, 290304, 1254400, 4465125, 18144000, 72990720, 391910400, 1881169920, 9754214400, 45660160000, 205752960000, 905748480000, 5280992640000, 28326238617600, 162956344320000, 853298675712000, 5309413982208000

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), May 25 2003

Keywords

• nonn

Comments

Minimum product of hook lengths of a partition of n. - Eric M. Schmidt, May 07 2013

Examples

			For n=4, we can have
abcd, abc and ab (the rest are symmetric).
......d.......cd
The hook products are 4! = 24, 4*2*1*1 = 8 and 3*2*2*1 = 12, so a(4) = 8. - _Jon Perry_
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..80
• Eric Weisstein's World of Mathematics, Hook Length Formula

Crossrefs

Cf. A003040.

Programs

Maple

# Maple code from  Emeric Deutsch, May 12 2004 (Start)
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:
seq(sort([seq(H(rev(partition(j)[i])),i=1..numbpart(j))])[1],j=1..30);
# the procedure H gives the hook product for a given partition written with parts in nonincreasing order;
# 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.
# (End)

Extensions

More terms from Eric M. Schmidt, May 07 2013