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Revision History for A082914 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

Showing entries 1-10 | older changes
A082914
a(n) = n! / A003040(n).
(history; published version)
#16 by N. J. A. Sloane at Thu Aug 17 22:20:29 EDT 2017
STATUS

editing

approved

#15 by N. J. A. Sloane at Thu Aug 17 22:20:07 EDT 2017
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: 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. # Emeric Deutsch, May 12 2004

# 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)

STATUS

approved

editing

Discussion
Thu Aug 17
22:20
N. J. A. Sloane: I redid the layout of the Maple section
#14 by N. J. A. Sloane at Thu Aug 17 22:16:58 EDT 2017
STATUS

proposed

approved

#13 by Michel Marcus at Mon Aug 14 00:53:30 EDT 2017
STATUS

editing

proposed

Discussion
Mon Aug 14
01:04
Jon E. Schoenfield: @Michel -- I've seen a number of places where users who signed a Maple contribution would do so with the format "# _Registered_Username_, Jan 01 2017" except when the signature itself followed something else that was a comment, in which case a hyphen was used instead of a "#".  Is it considered better to use another "#" immediately before the " _Registered User_, Jan 01 2017"?
01:05
Jon E. Schoenfield: (If that question didn't make sense, I'll be glad to try to state it more clearly!)  :-)
Wed Aug 16
00:08
Jon E. Schoenfield: (I tried asking the question a different way at the draft of A095149.)
23:33
Jon E. Schoenfield: Given Alois's pink-box comment and edit at A095149, I'm wondering whether

     # 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. # _Emeric Deutsch_, May 12 2004 

should instead be

     # 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. - _Emeric Deutsch_, May 12 2004

Is it worth changing?
Thu Aug 17
22:16
N. J. A. Sloane: I am going to approve this so that I can see it w/o all the underscores, and then I will edit it
#12 by Michel Marcus at Mon Aug 14 00:52:51 EDT 2017
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: 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. - _# _Emeric Deutsch_, May 12 2004

STATUS

proposed

editing

Discussion
Mon Aug 14
00:53
Michel Marcus: coming from A092031
#11 by Jon E. Schoenfield at Sun Aug 13 20:33:28 EDT 2017
STATUS

editing

proposed

#10 by Jon E. Schoenfield at Sun Aug 13 20:33:06 EDT 2017
EXAMPLE

For n=4, we can have;

The hook products are 4! = 24, 4.*2.*1.*1 = 8 and 3.*2.*2.*1 = 12, so a(4) = 8. (_- _Jon Perry_)

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: 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. (- _Emeric Deutsch)_

STATUS

approved

editing

#9 by Ralf Stephan at Sun Oct 20 02:47:32 EDT 2013
STATUS

editing

approved

#8 by Ralf Stephan at Sun Oct 20 02:47:18 EDT 2013
LINKS

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HookLengthFormula.html">Hook Length Formula</a>

EXAMPLE

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)

STATUS

proposed

editing

#7 by Arkadiusz Wesolowski at Sat Oct 19 17:22:25 EDT 2013
STATUS

editing

proposed

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