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The number of unrestricted partitions of the non-negative integer n.
The integer n must be small.
The unrestricted partitions of the positive integer n.
This function returns a sequence of integer sequences, each
of which is a different sequence of positive integers (in descending
order) adding up to n.
The integer n must be small.
The collection of all partitions of the positive integer n into
k parts. This function returns a sequence of integer sequences, each
of which is a distinct sequence of positive integers (in descending
order) whose sum is n.
The partitions of the positive integer n, where the parts are restricted
to being elements of the set M, which must contain positive integers only.
This function returns a sequence of integer sequences, each
of which is a different sequence of positive integers (in descending
order) whose sum is n and each contained in M (repetitions are
allowed in the partition).
The partitions of the positive integer n into k parts, where the parts
are restricted to being elements of the set M, which must contain positive
integers only. This function returns a sequence of integer sequences, each
of which is a different sequence of positive integers (in descending
order) whose sum is n and each contained in M (repetitions are
allowed in the partition).
A partition is considered to be a sequence of weakly decreasing positive
integers. A sequence is allowed to have trailing zeros, and the empty sequence
is accepted as a partition (of zero).
True if the young diagram of P1 covers the young diagram of P2.
(see below for description of young diagram).
For the young diagram corresponding to partition P, returned is the
partition corresponding to its conjugate diagram.
(see below for description of young diagram).
Will return the integer which the partition P is a partition of. Which
is simply the sum of its entries.
Lexicographical ordering of partitions is such that for partitions P1 and P2,
P1 > P2 implies that P1 is greater in the first non-equal entry.
The function returns the index of P with respect to lexicographical ordering of
all the partitions of the same weight. The first index is zero.
The conjugacy classes of the symmetric group on n elements correspond to
the partitions of n. The function PartnToElt below converts a
partition to an element of the corresponding conjugacy class.
> PartitionToElt := function(G, p)
> x := [];
> s := 0;
> for d in p do
> x cat:= Rotate([s+1 .. s+d], -1);
> s +:= d;
> end for;
> return G!x;
> end function;
>
> ConjClasses := function(n)
> G := Sym(n);
> return [ PartitionToElt(G, p) : p in Partitions(n) ];
> end function;
>
> ConjClasses(5);
[
(1, 2, 3, 4, 5),
(1, 2, 3, 4),
(1, 2, 3)(4, 5),
(1, 2, 3),
(1, 2)(3, 4),
(1, 2),
Id($)
]
> Classes(Sym(5));
Conjugacy Classes
-----------------
[1] Order 1 Length 1
Rep Id($)
[2] Order 2 Length 10
Rep (1, 2)
[3] Order 2 Length 15
Rep (1, 2)(3, 4)
[4] Order 3 Length 20
Rep (1, 2, 3)
[5] Order 4 Length 30
Rep (1, 2, 3, 4)
[6] Order 5 Length 24
Rep (1, 2, 3, 4, 5)
[7] Order 6 Length 20
Rep (1, 2, 3)(4, 5)
The number of ways of changing one dollar into five, ten, twenty and fifty
cent coins can be calculated using RestrictedPartitions. There is also
a well known solution from generating functions, which we use as a check.
> coins := {5, 10, 20, 50};
> T := [#RestrictedPartitions(n, coins) : n in [0 .. 100 by 5]];
> T;
[ 1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 13, 13, 18, 18, 24, 24, 31, 31, 39, 39, 49 ]
> F<t> := PowerSeriesRing(RationalField(), 101);
> &*[1/(1-t^i) : i in coins];
1 + t^5 + 2*t^10 + 2*t^15 + 4*t^20 + 4*t^25 + 6*t^30 + 6*t^35 + 9*t^40 + 9*t^45
+ 13*t^50 + 13*t^55 + 18*t^60 + 18*t^65 + 24*t^70 + 24*t^75 + 31*t^80 +
31*t^85 + 39*t^90 + 39*t^95 + 49*t^100 + O(t^101)
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