

A187449


Number of ways to choose three elements from the multiset representative corresponding to the kth multiset repetition class defining partition of n in AbramowitzStegun order.


3



0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 3, 2, 1, 4, 3, 2, 1, 5, 4, 4, 3, 2, 1, 4, 6, 4, 4, 3, 2, 1, 7, 6, 4, 4, 3, 2, 1, 8, 7, 7, 6, 4, 4, 4, 3, 2, 1, 10, 8, 8, 7, 6, 4, 4, 4, 3, 2, 1, 11, 8, 8, 7, 6, 4, 4, 4, 3, 2, 1, 10, 11, 9, 8, 8, 7, 7, 4, 6, 4, 4, 4, 3, 2, 1, 14, 13, 12, 11, 9, 8, 8, 7, 7, 4, 6, 4, 4, 4, 3, 2, 1, 15, 14, 12, 11, 9, 8, 8, 7, 7, 4, 6, 4, 4, 4, 3, 2, 1, 18, 15, 14, 10, 12, 9, 11, 9, 7, 8, 8, 7, 4, 7, 4, 6, 4, 4, 4, 3, 2, 1, 19, 15, 15, 14, 12, 10, 12, 9, 11, 9, 7, 8, 8, 7, 4, 7, 4, 6, 4, 4, 4, 3, 2, 1, 16, 19, 15, 15, 14, 12, 10, 12, 9, 11, 9, 7, 8, 8, 7, 4, 7, 4, 6, 4, 4, 4, 3, 2, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,8


COMMENTS

This array a(n,k), called (Multiset choose 3), is also denoted by MS(3;n,k). It is the third member of an lfamily of arrays for multiset choose l, called MS(l;n,k).
The length of row n of this array is A007294(n).
MS(3;n,k) gives the number of ways to choose three elements from the multiset encoded by the kth multiset repetition class defining partition of n. The AbramowitzStegun (ASt) order for partitions is used (see A036036). For the characteristic array of multiset repetition class defining partitions of n in ASt order see A176723.
Note that multiset choose l should not be confused with N multichoose l (see, e.g., Wolfram's Mathworld). Here one picks from a multiset l=3 of its elements.
Ordinary sets appear exactly for n=T(N), with the triangular numbers T(N):=A000217(N). They are defined by the partition 1,2,...,N, and a(T(N),1)= binomial(N,3). For n=0 the empty multiset appears.


REFERENCES

See A176725.


LINKS

Table of n, a(n) for n=0..192.


FORMULA

a(n,k) gives the number of ways to choose three elements from any multiset of the repetition class defined by the kth multiset repetition class defining partition of n, with k=1,...,A007294(n).
a(n,k) = binomial(MS(1;n,k)+2,3)  (m(n,k)[2] + MS(1;n,k)*m(n,k)[1]), with MS(1;n,k):=A176725(n,k), and m(n,k)[j] the jth member of the multiplicity list m(n,k) of the exponents of the kth multiset repetition class defining partition of n.


EXAMPLE

[0],
[0],
[0],
[0,1],
[1,1],
[2,1],
[1,2,2,1],
[3,3,2,1],
[4,3,2,1],
[5,4,4,3,2,1],
[4,6,4,4,3,2,1],
[7,6,4,4,3,2,1],
...
a(6,2)=MS(3;6,2)=2 because the second multiset repetition class defining partition of n=6 is [1^2,2^2] (from the characteristic array A176723) encoding the 4multiset representative {1,1,2,2}, and there are 2 possibilities to choose 3 elements from this set, namely 1,1,2 and 1,2,2.
a(T(4),1)=a(10,1)=MS(3;10,1) = 4 from the ordinary set {1,2,3,4} (defined by the multiset repetition class defining partition 1,2,3,4). This coincides with binomial(4,3)=4.
a(7,3)=MS(3;7,3)=2 from {1,1,1,1,1,2} choose 3 which is 2, namely 1,1,1 and 1,1,2. The corresponding multiset repetition class defining partition has exponent list [5,1] and this is the 15th in the ASt ordered list of such partitions.
a(8,2) = 3 = binomial(2+2,3)  (1+2*0), because the relevant partition has exponents [4,2] (corresponding to the 6multiset representative {1,1,1,1,2,2}) with MS(1;8,2)= 2, and the multiplicity list of the exponents is m(8,2)=[0,1,0,1]. Hence m(8,2)[2]=1 and m(8,2)[1]=0. The three choices are 1,1,1; 1,1,2 and 1,2,2.
a(10,3)= 4 = binomial(3+2,2)  (0+3*2)= 106, corresponding to the 7multiset representative {1,1,1,1,1,2,3} with MS(1;10,3)= 3, exponents [5,1,1] with multiplicity list m(10,3)=[2,0,0,0,1]. Hence m(10,3)[2]=0 and m(10,3)[1]=2. The four choices are 1,1,1; 1,1,2; 1,1,3 and 1,2,3.


CROSSREFS

Cf. A176725: MS(1;n,k), A187445: MS(2;n,k).
Sequence in context: A008612 A029320 A187450 * A102541 A243928 A286363
Adjacent sequences: A187446 A187447 A187448 * A187450 A187451 A187452


KEYWORD

nonn,tabf


AUTHOR

Wolfdieter Lang, Mar 15 2011


EXTENSIONS

Changed (in response to comments from Franklin T. AdamsWatters) by Wolfdieter Lang, Apr 02 2011


STATUS

approved



