Dawn Endico writes in his e-mail of 3-Dec-92:
Some of you may be interested to know that I was able to compile GAP
on an Amdahl mainframe under UTS unix. (System V.3.1 / UTS 2.1)
I did it using "make usg".I was wondering if some sort of standard test routines already exist
that I could use to exercise this new system. I would want to check
for both correct results as well as timing. It would be useful to
compare performance on this machine to implementations on other
systems.
We don't have a complete test set. The following file 'combinat.tst' is
our standard test. Start GAP and issue the command 'ReadTest(
"combinat.tst" );'. This will read the file and compare the output with
the comments (those beginning with '#>') in this file. If they match,
nothing is printed, otherwise the differences are printed. So if you see
gap> ReadTest( "combinat.tst" ); combinat 3.1 1992/04/29 50000 GAPstones gap>
you know that everything worked as expected. 'combinat.tst' exercises
GAP quite a bit, with lots of function calls, several garbage
collections, lots of list manipulations, etc. So if 'combinat.tst'
passes it is likely that everything else will also work.
GAPstones are used to measure GAP's performance. An Atari ST achieves
about 1000 GAPstones (actually now with GNU cc 2.2 it is more like 1500),
our DECstations 5120 achieve about 22000 GAPstones, and our HP 9000/730
achieves about 65000 GAPstones. This is currently the best result I am
aware of. Actually I would be quite interested in further results. So
if you have another machine, run this test and e-mail me the results, I
will summarize them in this forum.
Martin.
#############################################################################
##
#A combinat.tst GAP tests Martin Schoenert
##
#A @(#)$Id: 1.html,v 1.2 2004/04/21 15:06:22 felsch Exp $
##
#Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file tests the functions that mainly deal with combinatorics.
##
#H $Log: 1.html,v $
#H Revision 1.2 2004/04/21 15:06:22 felsch
#H Corrected links in the Forum Archive pages. VF
#H
#H Revision 1.1.1.1 2004/04/20 13:39:30 felsch
#H The final GAP-Forum archive until 2003.
#H
#H Revision 1.4 2003/06/12 19:20:32 gap
#H Further update. AH
#H
#H Revision 1.3 1997/08/15 11:18:49 gap
#H New forum setup. AH
#H
#H Revision 1.2 1997/04/24 15:24:52 gap
#H These files were replaced by the versions in WWW. The content is basically the
#H same but the formatting has been much more friendly towards the HTML-Converter.
#H AH
#H
#H Revision 1.1 1996/10/30 13:06:34 gap
#H added forum archive and translation files.
#H
#H Revision 3.1 1992/04/29 08:53:40 martin
#H changed the output (because of the new printer)
#H
#H Revision 3.0 1991/07/22 18:13:03 martin
#H initial revision under RCS
#H
##
#F Factorial( <n> ) . . . . . . . . . . . . . . . . factorial of an integer
List( [0..10], Factorial );
#>[ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ]
Factorial( 50 );
#>30414093201713378043612608166064768844377641568960512000000000000
#F Binomial( <n>, <k> ) . . . . . . . . . binomial coefficient of integers
List( [-8..8], k -> Binomial( 0, k ) );
#>[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
List( [-8..8], n -> Binomial( n, 0 ) );
#>[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
ForAll( [-8..8], n -> ForAll( [-2..8], k ->
Binomial(n,k) = Binomial(n-1,k) + Binomial(n-1,k-1) ) );
#>true
Binomial( 400, 50 );
#>17035900270730601418919867558071677342938596450600561760371485120
#F Bell( <n> ) . . . . . . . . . . . . . . . . . value of the Bell sequence
List( [0..10], n -> Bell(n) );
#>[ 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975 ]
List( [0..10], n -> Sum( [0..n], k -> Stirling2( n, k ) ) );
#>[ 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975 ]
Bell( 60 );
#>976939307467007552986994066961675455550246347757474482558637
#F Stirling1( <n>, <k> ) . . . . . . . . . Stirling number of the first kind
List( [-8..8], k -> Stirling1( 0, k ) );
#>[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
List( [-8..8], n -> Stirling1( n, 0 ) );
#>[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
ForAll( [-8..8], n -> ForAll( [-8..8], k ->
Stirling1(n,k) = (n-1) * Stirling1(n-1,k) + Stirling1(n-1,k-1) ) );
#>true
Stirling1( 60, 20 );
#>568611292461582075463109862277030309493811818619783570055397018154658816
#F Stirling2( <n>, <k> ) . . . . . . . . Stirling number of the second kind
List( [-8..8], k -> Stirling2( 0, k ) );
#>[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
List( [-8..8], n -> Stirling2( n, 0 ) );
#>[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
ForAll( [-8..8], n -> ForAll( [-8..8], k ->
Stirling2(n,k) = k * Stirling2(n-1,k) + Stirling2(n-1,k-1) ) );
#>true
Stirling2( 60, 20 );
#>170886257768137628374668205554120607567311094075812403938286
#F Combinations( <mset>, <k> ) . . . . set of sorted sublists of a multiset
Combinations( [] );
#>[ [ ] ]
List( [0..1], k -> Combinations( [], k ) );
#>[ [ [ ] ], [ ] ]
Combinations( [1..4] );
#>[ [ ], [ 1 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 2, 3, 4 ], [ 1, 2, 4 ], [ 1, 3 ],
#> [ 1, 3, 4 ], [ 1, 4 ], [ 2 ], [ 2, 3 ], [ 2, 3, 4 ], [ 2, 4 ], [ 3 ],
#> [ 3, 4 ], [ 4 ] ]
List( [0..5], k -> Combinations( [1..4], k ) );
#>[ [ [ ] ], [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ],
#> [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ],
#> [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], [ 2, 3, 4 ] ], [ [ 1, 2, 3, 4 ] ],
#> [ ] ]
Combinations( [1,2,2,3] );
#>[ [ ], [ 1 ], [ 1, 2 ], [ 1, 2, 2 ], [ 1, 2, 2, 3 ], [ 1, 2, 3 ], [ 1, 3 ],
#> [ 2 ], [ 2, 2 ], [ 2, 2, 3 ], [ 2, 3 ], [ 3 ] ]
List( [0..5], k -> Combinations( [1,2,2,3], k ) );
#>[ [ [ ] ], [ [ 1 ], [ 2 ], [ 3 ] ],
#> [ [ 1, 2 ], [ 1, 3 ], [ 2, 2 ], [ 2, 3 ] ],
#> [ [ 1, 2, 2 ], [ 1, 2, 3 ], [ 2, 2, 3 ] ], [ [ 1, 2, 2, 3 ] ], [ ] ]
Combinations( [1..12] )[4039];
#>[ 7, 8, 9, 10, 11, 12 ]
Combinations( [1..16], 4 )[266];
#>[ 1, 5, 9, 13 ]
Combinations( [1,2,3,3,4,4,5,5,5,6,6,6,7,7,7,7] )[378];
#>[ 1, 2, 3, 4, 5, 6, 7 ]
Combinations( [1,2,3,3,4,4,5,5,5,6,6,6,7,7,7,7,8,8,8,8], 8 )[97];
#>[ 1, 2, 3, 4, 5, 6, 7, 8 ]
#F NrCombinations( <mset>, <k> ) . . number of sorted sublists of a multiset
NrCombinations( [] );
#>1
List( [0..1], k -> NrCombinations( [], k ) );
#>[ 1, 0 ]
NrCombinations( [1..4] );
#>16
List( [0..5], k -> NrCombinations( [1..4], k ) );
#>[ 1, 4, 6, 4, 1, 0 ]
NrCombinations( [1,2,2,3] );
#>12
List( [0..5], k -> NrCombinations( [1,2,2,3], k ) );
#>[ 1, 3, 4, 3, 1, 0 ]
NrCombinations( [1..12] );
#>4096
NrCombinations( [1..16], 4 );
#>1820
NrCombinations( [1,2,3,3,4,4,5,5,5,6,6,6,7,7,7,7] );
#>2880
NrCombinations( [1,2,3,3,4,4,5,5,5,6,6,6,7,7,7,7,8,8,8,8], 8 );
#>1558
#F Arrangements( <mset> ) . . . . set of ordered combinations of a multiset
Arrangements( [] );
#>[ [ ] ]
List( [0..1], k -> Arrangements( [], k ) );
#>[ [ [ ] ], [ ] ]
Arrangements( [1..3] );
#>[ [ ], [ 1 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 3 ], [ 1, 3, 2 ], [ 2 ], [ 2, 1 ],
#> [ 2, 1, 3 ], [ 2, 3 ], [ 2, 3, 1 ], [ 3 ], [ 3, 1 ], [ 3, 1, 2 ], [ 3, 2 ],
#> [ 3, 2, 1 ] ]
List( [0..4], k -> Arrangements( [1..3], k ) );
#>[ [ [ ] ], [ [ 1 ], [ 2 ], [ 3 ] ],
#> [ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ] ],
#> [ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 2, 1, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ],
#> [ 3, 2, 1 ] ], [ ] ]
Arrangements( [1,2,2,3] );
#>[ [ ], [ 1 ], [ 1, 2 ], [ 1, 2, 2 ], [ 1, 2, 2, 3 ], [ 1, 2, 3 ],
#> [ 1, 2, 3, 2 ], [ 1, 3 ], [ 1, 3, 2 ], [ 1, 3, 2, 2 ], [ 2 ], [ 2, 1 ],
#> [ 2, 1, 2 ], [ 2, 1, 2, 3 ], [ 2, 1, 3 ], [ 2, 1, 3, 2 ], [ 2, 2 ],
#> [ 2, 2, 1 ], [ 2, 2, 1, 3 ], [ 2, 2, 3 ], [ 2, 2, 3, 1 ], [ 2, 3 ],
#> [ 2, 3, 1 ], [ 2, 3, 1, 2 ], [ 2, 3, 2 ], [ 2, 3, 2, 1 ], [ 3 ], [ 3, 1 ],
#> [ 3, 1, 2 ], [ 3, 1, 2, 2 ], [ 3, 2 ], [ 3, 2, 1 ], [ 3, 2, 1, 2 ],
#> [ 3, 2, 2 ], [ 3, 2, 2, 1 ] ]
List( [0..5], k -> Arrangements( [1,2,2,3], k ) );
#>[ [ [ ] ], [ [ 1 ], [ 2 ], [ 3 ] ],
#> [ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ], [ 3, 2 ] ],
#> [ [ 1, 2, 2 ], [ 1, 2, 3 ], [ 1, 3, 2 ], [ 2, 1, 2 ], [ 2, 1, 3 ],
#> [ 2, 2, 1 ], [ 2, 2, 3 ], [ 2, 3, 1 ], [ 2, 3, 2 ], [ 3, 1, 2 ],
#> [ 3, 2, 1 ], [ 3, 2, 2 ] ],
#> [ [ 1, 2, 2, 3 ], [ 1, 2, 3, 2 ], [ 1, 3, 2, 2 ], [ 2, 1, 2, 3 ],
#> [ 2, 1, 3, 2 ], [ 2, 2, 1, 3 ], [ 2, 2, 3, 1 ], [ 2, 3, 1, 2 ],
#> [ 2, 3, 2, 1 ], [ 3, 1, 2, 2 ], [ 3, 2, 1, 2 ], [ 3, 2, 2, 1 ] ], [ ] ]
Arrangements( [1..6] )[736];
#>[ 3, 2, 1, 6, 5, 4 ]
Arrangements( [1..8], 4 )[443];
#>[ 3, 1, 7, 5 ]
Arrangements( [1,2,3,3,4,4,5] )[3511];
#>[ 5, 4, 3, 2, 1 ]
Arrangements( [1,2,3,4,4,5,5,6,6], 5 )[424];
#>[ 2, 3, 4, 5, 6 ]
#F NrArrangements( <mset>, <k> ) . . number of sorted sublists of a multiset
NrArrangements( [] );
#>1
List( [0..1], k -> NrArrangements( [], k ) );
#>[ 1, 0 ]
NrArrangements( [1..3] );
#>16
List( [0..4], k -> NrArrangements( [1..3], k ) );
#>[ 1, 3, 6, 6, 0 ]
NrArrangements( [1,2,2,3] );
#>35
List( [0..5], k -> NrArrangements( [1,2,2,3], k ) );
#>[ 1, 3, 7, 12, 12, 0 ]
NrArrangements( [1..6] );
#>1957
NrArrangements( [1..8], 4 );
#>1680
NrArrangements( [1,2,3,3,4,4,5] );
#>3592
NrArrangements( [1,2,3,4,4,5,5,6,6], 5 );
#>2880
#F UnorderedTuples( <set>, <k> ) . . . . set of unordered tuples from a set
List( [0..1], k -> UnorderedTuples( [], k ) );
#>[ [ [ ] ], [ ] ]
List( [0..4], k -> UnorderedTuples( [1..3], k ) );
#>[ [ [ ] ], [ [ 1 ], [ 2 ], [ 3 ] ],
#> [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 2 ], [ 2, 3 ], [ 3, 3 ] ],
#> [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 1, 1, 3 ], [ 1, 2, 2 ], [ 1, 2, 3 ],
#> [ 1, 3, 3 ], [ 2, 2, 2 ], [ 2, 2, 3 ], [ 2, 3, 3 ], [ 3, 3, 3 ] ],
#> [ [ 1, 1, 1, 1 ], [ 1, 1, 1, 2 ], [ 1, 1, 1, 3 ], [ 1, 1, 2, 2 ],
#> [ 1, 1, 2, 3 ], [ 1, 1, 3, 3 ], [ 1, 2, 2, 2 ], [ 1, 2, 2, 3 ],
#> [ 1, 2, 3, 3 ], [ 1, 3, 3, 3 ], [ 2, 2, 2, 2 ], [ 2, 2, 2, 3 ],
#> [ 2, 2, 3, 3 ], [ 2, 3, 3, 3 ], [ 3, 3, 3, 3 ] ] ]
UnorderedTuples( [1..10], 6 )[1459];
#>[ 1, 3, 5, 7, 9, 10 ]
#F NrUnorderedTuples( <set>, <k> ) . . number unordered of tuples from a set
List( [0..1], k -> NrUnorderedTuples( [], k ) );
#>[ 1, 0 ]
List( [0..4], k -> NrUnorderedTuples( [1..3], k ) );
#>[ 1, 3, 6, 10, 15 ]
NrUnorderedTuples( [1..10], 6 );
#>5005
#F Tuples( <set>, <k> ) . . . . . . . . . set of ordered tuples from a set
List( [0..1], k -> Tuples( [], k ) );
#>[ [ [ ] ], [ ] ]
List( [0..3], k -> Tuples( [1..3], k ) );
#>[ [ [ ] ], [ [ 1 ], [ 2 ], [ 3 ] ],
#> [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ],
#> [ 3, 2 ], [ 3, 3 ] ],
#> [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 1, 1, 3 ], [ 1, 2, 1 ], [ 1, 2, 2 ],
#> [ 1, 2, 3 ], [ 1, 3, 1 ], [ 1, 3, 2 ], [ 1, 3, 3 ], [ 2, 1, 1 ],
#> [ 2, 1, 2 ], [ 2, 1, 3 ], [ 2, 2, 1 ], [ 2, 2, 2 ], [ 2, 2, 3 ],
#> [ 2, 3, 1 ], [ 2, 3, 2 ], [ 2, 3, 3 ], [ 3, 1, 1 ], [ 3, 1, 2 ],
#> [ 3, 1, 3 ], [ 3, 2, 1 ], [ 3, 2, 2 ], [ 3, 2, 3 ], [ 3, 3, 1 ],
#> [ 3, 3, 2 ], [ 3, 3, 3 ] ] ]
Tuples( [1..8], 4 )[167];
#>[ 1, 3, 5, 7 ]
#F NrTuples( <set>, <k> ) . . . . . . . number of ordered tuples from a set
List( [0..1], k -> NrTuples( [], k ) );
#>[ 1, 0 ]
List( [0..3], k -> NrTuples( [1..3], k ) );
#>[ 1, 3, 9, 27 ]
NrTuples( [1..8], 4 );
#>4096
#F PermutationsList( <mset> ) . . . . . . set of permutations of a multiset
PermutationsList( [] );
#>[ [ ] ]
PermutationsList( [1..4] );
#>[ [ 1, 2, 3, 4 ], [ 1, 2, 4, 3 ], [ 1, 3, 2, 4 ], [ 1, 3, 4, 2 ],
#> [ 1, 4, 2, 3 ], [ 1, 4, 3, 2 ], [ 2, 1, 3, 4 ], [ 2, 1, 4, 3 ],
#> [ 2, 3, 1, 4 ], [ 2, 3, 4, 1 ], [ 2, 4, 1, 3 ], [ 2, 4, 3, 1 ],
#> [ 3, 1, 2, 4 ], [ 3, 1, 4, 2 ], [ 3, 2, 1, 4 ], [ 3, 2, 4, 1 ],
#> [ 3, 4, 1, 2 ], [ 3, 4, 2, 1 ], [ 4, 1, 2, 3 ], [ 4, 1, 3, 2 ],
#> [ 4, 2, 1, 3 ], [ 4, 2, 3, 1 ], [ 4, 3, 1, 2 ], [ 4, 3, 2, 1 ] ]
PermutationsList( [1,2,2,3,] );
#>[ [ 1, 2, 2, 3 ], [ 1, 2, 3, 2 ], [ 1, 3, 2, 2 ], [ 2, 1, 2, 3 ],
#> [ 2, 1, 3, 2 ], [ 2, 2, 1, 3 ], [ 2, 2, 3, 1 ], [ 2, 3, 1, 2 ],
#> [ 2, 3, 2, 1 ], [ 3, 1, 2, 2 ], [ 3, 2, 1, 2 ], [ 3, 2, 2, 1 ] ]
PermutationsList( [1..6] )[ 128 ];
#>[ 2, 1, 4, 3, 6, 5 ]
PermutationsList( [1,2,2,3,3,4,4,4] )[1359];
#>[ 4, 3, 2, 1, 4, 3, 2, 4 ]
#F NrPermutationsList( <mset> ) . . . number of permutations of a multiset
NrPermutationsList( [] );
#>1
NrPermutationsList( [1..4] );
#>24
NrPermutationsList( [1,2,2,3] );
#>12
NrPermutationsList( [1..6] );
#>720
NrPermutationsList( [1,2,2,3,3,4,4,4] );
#>1680
#F Derangements( <list> ) . . . . set of fixpointfree permutations of a list
Derangements( [] );
#>[ [ ] ]
Derangements( [1..4] );
#>[ [ 2, 1, 4, 3 ], [ 2, 3, 4, 1 ], [ 2, 4, 1, 3 ], [ 3, 1, 4, 2 ],
#> [ 3, 4, 1, 2 ], [ 3, 4, 2, 1 ], [ 4, 1, 2, 3 ], [ 4, 3, 1, 2 ],
#> [ 4, 3, 2, 1 ] ]
Derangements( [1..6] )[ 128 ];
#>[ 4, 3, 6, 1, 2, 5 ]
Derangements( [1,2,2,3,3,4,4,4] )[64];
#>[ 4, 1, 4, 2, 4, 2, 3, 3 ]
#F NrDerangements( <list> ) . number of fixpointfree permutations of a list
NrDerangements( [] );
#>1
NrDerangements( [1..4] );
#>9
NrDerangements( [1..6] );
#>265
NrDerangements( [1,2,2,3,3,4,4,4] );
#>126
#F Permanent( <mat> ) . . . . . . . . . . . . . . . . permanent of a matrix
Permanent( [[0,1,1,1],[1,0,1,1],[1,1,0,1],[1,1,1,0]] );
#>9
Permanent( [[1,1,0,1,0,0,0],[0,1,1,0,1,0,0],[0,0,1,1,0,1,0],[0,0,0,1,1,0,1],
[1,0,0,0,1,1,0],[0,1,0,0,0,1,1],[1,0,1,0,0,0,1]] );
#>24
#F PartitionsSet( <set> ) . . . . . . . . . . . set of partitions of a set
PartitionsSet( [] );
#>[ [ ] ]
List( [0..1], k -> PartitionsSet( [], k ) );
#>[ [ [ ] ], [ ] ]
PartitionsSet( [1..4] );
#>[ [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ], [ [ 1 ], [ 2 ], [ 3, 4 ] ],
#> [ [ 1 ], [ 2, 3 ], [ 4 ] ], [ [ 1 ], [ 2, 3, 4 ] ],
#> [ [ 1 ], [ 2, 4 ], [ 3 ] ], [ [ 1, 2 ], [ 3 ], [ 4 ] ],
#> [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 2, 3 ], [ 4 ] ], [ [ 1, 2, 3, 4 ] ],
#> [ [ 1, 2, 4 ], [ 3 ] ], [ [ 1, 3 ], [ 2 ], [ 4 ] ], [ [ 1, 3 ], [ 2, 4 ] ],
#> [ [ 1, 3, 4 ], [ 2 ] ], [ [ 1, 4 ], [ 2 ], [ 3 ] ], [ [ 1, 4 ], [ 2, 3 ] ] ]
List( [0..4], k -> PartitionsSet( [1..3], k ) );
#>[ [ ], [ [ [ 1, 2, 3 ] ] ],
#> [ [ [ 1 ], [ 2, 3 ] ], [ [ 1, 2 ], [ 3 ] ], [ [ 1, 3 ], [ 2 ] ] ],
#> [ [ [ 1 ], [ 2 ], [ 3 ] ] ], [ ] ]
PartitionsSet( [1..7] )[521];
#>[ [ 1, 3, 5, 7 ], [ 2, 4, 6 ] ]
PartitionsSet( [1..8], 3 )[96];
#>[ [ 1, 2, 3 ], [ 4, 5 ], [ 6, 7, 8 ] ]
#F NrPartitionsSet( <set> ) . . . . . . . . . number of partitions of a set
NrPartitionsSet( [] );
#>1
List( [0..1], k -> NrPartitionsSet( [], k ) );
#>[ 1, 0 ]
NrPartitionsSet( [1..4] );
#>15
List( [0..4], k -> NrPartitionsSet( [1,2,3], k ) );
#>[ 0, 1, 3, 1, 0 ]
NrPartitionsSet( [1..8] );
#>4140
NrPartitionsSet( [1..9], 3 );
#>3025
#F Partitions( <n> ) . . . . . . . . . . . . set of partitions of an integer
Partitions( 0 );
#>[ [ ] ]
List( [0..1], k -> Partitions( 0, k ) );
#>[ [ [ ] ], [ ] ]
Partitions( 6 );
#>[ [ 1, 1, 1, 1, 1, 1 ], [ 2, 1, 1, 1, 1 ], [ 2, 2, 1, 1 ], [ 2, 2, 2 ],
#> [ 3, 1, 1, 1 ], [ 3, 2, 1 ], [ 3, 3 ], [ 4, 1, 1 ], [ 4, 2 ], [ 5, 1 ],
#> [ 6 ] ]
List( [0..7], k -> Partitions( 6, k ) );
#>[ [ ], [ [ 6 ] ], [ [ 3, 3 ], [ 4, 2 ], [ 5, 1 ] ],
#> [ [ 2, 2, 2 ], [ 3, 2, 1 ], [ 4, 1, 1 ] ],
#> [ [ 2, 2, 1, 1 ], [ 3, 1, 1, 1 ] ], [ [ 2, 1, 1, 1, 1 ] ],
#> [ [ 1, 1, 1, 1, 1, 1 ] ], [ ] ]
Partitions( 20 )[314];
#>[ 7, 4, 3, 3, 2, 1 ]
Partitions( 20, 10 )[17];
#>[ 5, 3, 3, 2, 2, 1, 1, 1, 1, 1 ]
#F NrPartitions( <n> ) . . . . . . . . . number of partitions of an integer
NrPartitions( 0 );
#>1
List( [0..1], k -> NrPartitions( 0, k ) );
#>[ 1, 0 ]
NrPartitions( 6 );
#>11
List( [0..7], k -> NrPartitions( 6, k ) );
#>[ 0, 1, 3, 3, 2, 1, 1, 0 ]
NrPartitions( 100 );
#>190569292
NrPartitions( 100, 10 );
#>2977866
#F OrderedPartitions( <n> ) . . . . set of ordered partitions of an integer
OrderedPartitions( 0 );
#>[ [ ] ]
List( [0..1], k -> OrderedPartitions( 0, k ) );
#>[ [ [ ] ], [ ] ]
OrderedPartitions( 5 );
#>[ [ 1, 1, 1, 1, 1 ], [ 1, 1, 1, 2 ], [ 1, 1, 2, 1 ], [ 1, 1, 3 ],
#> [ 1, 2, 1, 1 ], [ 1, 2, 2 ], [ 1, 3, 1 ], [ 1, 4 ], [ 2, 1, 1, 1 ],
#> [ 2, 1, 2 ], [ 2, 2, 1 ], [ 2, 3 ], [ 3, 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5 ] ]
List( [0..6], k -> OrderedPartitions( 5, k ) );
#>[ [ ], [ [ 5 ] ], [ [ 1, 4 ], [ 2, 3 ], [ 3, 2 ], [ 4, 1 ] ],
#> [ [ 1, 1, 3 ], [ 1, 2, 2 ], [ 1, 3, 1 ], [ 2, 1, 2 ], [ 2, 2, 1 ],
#> [ 3, 1, 1 ] ],
#> [ [ 1, 1, 1, 2 ], [ 1, 1, 2, 1 ], [ 1, 2, 1, 1 ], [ 2, 1, 1, 1 ] ],
#> [ [ 1, 1, 1, 1, 1 ] ], [ ] ]
OrderedPartitions( 13 )[2048];
#>[ 1, 12 ]
OrderedPartitions( 16, 6 )[1001];
#>[ 1, 11, 1, 1, 1, 1 ]
#F NrOrderedPartitions( <n> ) . . number of ordered partitions of an integer
NrOrderedPartitions( 0 );
#>1
List( [0..1], k -> NrOrderedPartitions( 0, k ) );
#>[ 1, 0 ]
NrOrderedPartitions( 5 );
#>16
List( [0..6], k -> NrOrderedPartitions( 5, k ) );
#>[ 0, 1, 4, 6, 4, 1, 0 ]
NrOrderedPartitions( 13 );
#>4096
NrOrderedPartitions( 16, 6 );
#>3003
#F RestrictedPartitions( <n>, <set> ) . restricted partitions of an integer
RestrictedPartitions( 0, [1..10] );
#>[ [ ] ]
List( [0..1], k -> RestrictedPartitions( 0, [1..10], k ) );
#>[ [ [ ] ], [ ] ]
RestrictedPartitions( 10, [1,2,5,10] );
#>[ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 2, 1, 1, 1, 1, 1, 1, 1, 1 ],
#> [ 2, 2, 1, 1, 1, 1, 1, 1 ], [ 2, 2, 2, 1, 1, 1, 1 ], [ 2, 2, 2, 2, 1, 1 ],
#> [ 2, 2, 2, 2, 2 ], [ 5, 1, 1, 1, 1, 1 ], [ 5, 2, 1, 1, 1 ], [ 5, 2, 2, 1 ],
#> [ 5, 5 ], [ 10 ] ]
List( [1..10], k -> RestrictedPartitions( 10, [1,2,5,10], k ) );
#>[ [ [ 10 ] ], [ [ 5, 5 ] ], [ ], [ [ 5, 2, 2, 1 ] ],
#> [ [ 2, 2, 2, 2, 2 ], [ 5, 2, 1, 1, 1 ] ],
#> [ [ 2, 2, 2, 2, 1, 1 ], [ 5, 1, 1, 1, 1, 1 ] ], [ [ 2, 2, 2, 1, 1, 1, 1 ] ],
#> [ [ 2, 2, 1, 1, 1, 1, 1, 1 ] ], [ [ 2, 1, 1, 1, 1, 1, 1, 1, 1 ] ],
#> [ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ] ]
RestrictedPartitions( 20, [2,5,10] );
#>[ [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ 5, 5, 2, 2, 2, 2, 2 ], [ 5, 5, 5, 5 ],
#> [ 10, 2, 2, 2, 2, 2 ], [ 10, 5, 5 ], [ 10, 10 ] ]
List( [1..20], k -> RestrictedPartitions( 20, [2,5,10], k ) );
#>[ [ ], [ [ 10, 10 ] ], [ [ 10, 5, 5 ] ], [ [ 5, 5, 5, 5 ] ], [ ],
#> [ [ 10, 2, 2, 2, 2, 2 ] ], [ [ 5, 5, 2, 2, 2, 2, 2 ] ], [ ], [ ],
#> [ [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] ], [ ], [ ], [ ], [ ], [ ], [ ],
#> [ ], [ ], [ ], [ ] ]
RestrictedPartitions( 60, [2,3,5,7,11,13,17] )[600];
#>[ 13, 7, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
RestrictedPartitions( 100, [2,3,5,7,11,13,17], 10 )[75];
#>[ 17, 17, 13, 13, 13, 7, 5, 5, 5, 5 ]
#F NrRestrictedPartitions(<n>,<set>) . . . . number of restricted partitions
NrRestrictedPartitions( 0, [1..10] );
#>1
List( [0..1], k -> NrRestrictedPartitions( 0, [1..10], k ) );
#>[ 1, 0 ]
NrRestrictedPartitions( 50, [1,2,5,10] );
#>341
List( [1..50], k -> NrRestrictedPartitions( 50, [1,2,5,10], k ) );
#>[ 0, 0, 0, 0, 1, 1, 1, 2, 4, 6, 6, 8, 10, 11, 11, 12, 13, 14, 14, 14, 15, 15,
#> 14, 14, 14, 13, 12, 12, 11, 10, 9, 9, 8, 7, 6, 6, 6, 5, 4, 4, 4, 3, 2, 2,
#> 2, 2, 1, 1, 1, 1 ]
NrRestrictedPartitions( 50, [2,5,10] );
#>21
List( [1..50], k -> NrRestrictedPartitions( 50, [2,5,10], k ) );
#>[ 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1,
#> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
NrRestrictedPartitions( 60, [2,3,5,7,11,13,17] );
#>1213
NrRestrictedPartitions( 100, [2,3,5,7,11,13,17], 10 );
#>125
#F Lucas(<P>,<Q>,<k>) . . . . . . . . . . . . . . value of a lucas sequence
List( [0..10], i->Lucas(1,-2,i)[1] );
#>[ 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341 ]
List( [0..10], i->Lucas(1,-2,i)[2] );
#>[ 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025 ]
List( [0..10], i->Lucas(1,-1,i)[1] );
#>[ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ]
List( [0..10], i->Lucas(2,1,i)[1] );
#>[ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
Lucas( 0, -4, 100 ) = [ 0, 2^101, 4^100 ];
#>true
#F Fibonacci( <n> ) . . . . . . . . . . . . value of the Fibonacci sequence
List( [0..17], Fibonacci );
#>[ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 ]
Fibonacci( 333 );
#>1751455877444438095408940282208383549115781784912085789506677971125378
#F Bernoulli( <n> ) . . . . . . . . . . . . value of the Bernoulli sequence
List( [0..14], Bernoulli );
#>[ 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6 ]
Bernoulli( 80 );
#>-4603784299479457646935574969019046849794257872751288919656867/230010
# thats it for the combinatorical package ##################################
Print("combinat 3.1 1992/04/29 ",QuoInt(700000000,time)," GAPstones\n");
if IsBound( GAPSTONES ) then Add( GAPSTONES, QuoInt(700000000,time) ); fi;