COMB PART,L / PART=PARTITIONS,<n>
lists all partitions of interger <n>.
COMB PART,L / PART=PARTITIONS,<n>,<m>
lists partitions of <n> consisting of <m> parts.
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In both forms of partitions the sizes of parts can be limited
by MIN and MAX specifications.
Example:
COMB PART1,CUR+1 / PART1=PARTITIONS,12,4 MIN=2 MAX=4
Partitions 4 of 12: N[PART1]=3
2 2 4 4
2 3 3 4
3 3 3 3
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By the DISTINCT specification only partitions with distinct parts
are accepted.
Example:
COMB PART1,CUR+1 / PART1=PARTITIONS,18,4 MIN=2 MAX=10 DISTINCT=1
Partitions 4 of 18: N[PART1]=5
2 3 4 9
2 3 5 8
2 3 6 7
2 4 5 7
3 4 5 6
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COMB P,L / P=PARTITIONS,<n>
PARTS=<ascending_list_of_positive_integers>
lists all partitions of <n> using only integers given by PARTS.
COMB P,L / PART=PARTITIONS,<n>,<m>
PARTS=<ascending_list_of_positive_integers>
lists partitions of <n> consisting of <m> parts using the PARTS integers.
As a special case, PARTS=POWERS,k is the same as
PARTS=1,2^k,3^k,4^k,...
Examples:
COMB P,CUR+1 / P=PARTITIONS,1729,2 PARTS=POWERS,3
Partitions 2 of 1729: N[P]=2
1 1728
729 1000
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PARTS=1,5,10,20,50,100,500,1000 (metal and paper moneys in Finland)
COMB P,CUR+1 / P=PARTITIONS,1000 RESULTS=0
Partitions of 1000: N[P]=2720784
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COMB P,L / P=PARTITIONS,<n>,<m> DISTINCT=1 OFF=i1,i2,...
makes partitions of size <m> for <n> with distinct elements and
excluding elements i1,i2,... listed by an OFF specification.
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Example:
COMB P,CUR+1 / P=PARTITIONS,25,3 DISTINCT=1 OFF=2,4,6,8,10
Partitions 3 of 25: N[P]=10
1 3 21
1 5 19
1 7 17
1 9 15
1 11 13
3 5 17
3 7 15
3 9 13
5 7 13
5 9 11
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COMB P,L / P=PARTITIONS,<n> GREATEST=<m>
lists partitions of <n> with <m> as the greatest part.
Example:
COMB P,CUR+1 / P=PARTITIONS,10 GREATEST=3
Partitions of 10: N[P]=8
1 3 3 3
2 2 3 3
1 1 2 3 3
1 2 2 2 3
1 1 1 1 3 3
1 1 1 2 2 3
1 1 1 1 1 2 3
1 1 1 1 1 1 1 3
The number of these partitions is the same as those obtained by
PARTITIONS,<n>,<m>
since there there is a bijective mapping between these two types of
partitions proved simply by transposing the Ferrers graph.
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COMB P,L / P=PARTITIONS,<n>,<m> MIN=<min> MAX=<max> MULTIN=1
both makes partitions of size <m> for <n> and - by assuming that each
partition f1+f2+...+fm=n represents frequencies of an n-fold trial with
m possible outcomes with equal probabilities 1/m - computes also the
probability that the conditions <min> <= fi <= <max>, i=1,2,...,m
are fulfilled. This probability is computed if MULTIN=1 is given.
Example on the next page!
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Example: What is the probability that in 600 tosses of an unbiased
dice the frequencies of each of the numbers 1,2,3,4,5,6 fall in
the closed interval [90,110] ?
TIME COUNT START
COMB P,CUR+2 / P=PARTITIONS,600,6 MIN=90 MAX=110 MULTIN=1 RESULTS=0
TIME COUNT END 0.220
Restricted partitions of 600: N[P]=5444 P=0.215947
Thus the probability is P=0.215947 .
Checking that the sum of all multinomial probabilities is 1:
Please note that it takes "some time". This is computed on a 700 MHz PC.
TIME COUNT START
COMB P,CUR+2 / P=PARTITIONS,600,6 MIN=0 MAX=600 MULTIN=1 RESULTS=0
TIME COUNT END 1406.632
Restricted partitions of 600: N[P]=981355696 P=1
C = Other forms of COMB
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