A187445 - OEIS

%I #15 Mar 30 2012 18:49:34

%S 0,0,1,1,1,2,1,2,1,3,3,2,1,4,3,2,1,4,3,2,1,5,4,3,3,2,1,6,5,4,3,3,2,1,

%T 7,5,4,3,3,2,1,7,6,5,5,3,4,3,3,2,1,8,7,6,5,5,3,4,3,3,2,1,8,7,6,5,5,3,

%U 4,3,3,2,1,10,8,6,7,6,5,5,3,5,3,4,3,3,2,1,11,9,8,8,6,7,6,5,5,3,5,3,4,3,3,2,1,11,9,8,8,6,7,6,5,5,3,5,3,4,3,3,2,1,12,11,9,6,8,6,8,6,5,7,6,5,3,5,3,5,3,4,3,3,2,1,12,11,9,9,8,6,8,6,8,6,5,7,6,5,3,5,3,5,3,4,3,3,2,1,10,12,11,9,9,8,6,8,6,8,6,5,7,6,5,3,5,3,5,3,4,3,3,2,1

%N Number of ways to choose two elements from the multiset representative corresponding to the k-th multiset repetition class defining partition of n in Abramowitz-Stegun order.

%C This array a(n,k), called (Multiset choose 2), is also denoted by MS(2;n,k).

%C MS(2;n,k) gives the number of ways to choose two elements from the multiset encoded by the k-th multiset repetition class defining partition of n. The Abramowitz-Stegun (A-St) order for partitions is used (see A036036).

%C The row lengths of this array are A007294(n).

%C For the characteristic array of multiset repetition class defining partitions of n in A-St order see A176723.

%C This is the second member of an l-family of arrays for multiset choose l, called MS(l;n,k).

%C For a list of the multiset repetition class defining partitions of n=1..15 see the link under A176725. For n=0 the empty partition appears.

%C Note that multiset choose k should not be confused with N multichoose k (see, e.g., Wolfram's Mathworld). Here one picks from a multiset k=2 of its elements.

%C Ordinary sets appear exactly for n=T(N), N>=1, with the triangular numbers T(N):=A000217(N). They are defined by the partitions 1,2,...,N, and a(T(N),1)= binomial(N,2).

%D See A176725.

%F a(n,k) gives the number of ways to choose two elements from the multiset defined by the k-th multiset defining partition of n, with k=1,...,A007294(n).

%F a(n,k) = binomial(MS(1;n,k)+1,2) - m(n,k)[1], with MS(1;n,k):=A176725(n,k), and m(n,k)[1] the first member of the multiplicity list m(n,k) of the exponents of the k-th multiset defining partition of n.

%e n=0..10:

%e [0],

%e [0],

%e [1],

%e [1, 1],

%e [2, 1],

%e [2, 1],

%e [3, 3, 2, 1],

%e [4, 3, 2, 1],

%e [4, 3, 2, 1],

%e [5, 4, 3, 3, 2, 1],

%e [6, 5, 4, 3, 3, 2, 1],

%e ...

%e a(6,2)=MS(2;6,2)=3 because the second multiset repetition class defining partition of n=6 is 1^2,2^2 (from the characteristic array A176723) encoding the 4-multiset representative {1,1,2,2}, and there are 3 ways to choose 2 elements from this set, namely 1,1; 2,2; and 1,2.

%e a(T(4),1)=a(10,1)=MS(2;10,1) = 6, from the ordinary set {1,2,3,4} (defined by the multiset repetition class defining partition 1,2,3,4), coinciding with binomial(4,2).

%e a(8,2)= 3 = binomial(2+1,2) - 0, because the relevant partition has exponents [4,2] (corresponding to the 6-multiset {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)[1]=0.

%e a(10,3)= 4 = binomial(3+1,2) - 2, corresponding to the 7-multiset {1,1,1,1,1,2,3} with MS(1;10,3)= 3 and exponents [5,1,1] with multiplicity m(10,3)=[2,0,0,0,1], hence m(10,3)[1]=2.

%Y Cf. A176725 MS(1;n,k), A187449 MS(3;n,k).

%K nonn,tabf

%O 0,6

%A _Wolfdieter Lang_, Mar 11 2011

%E Changed in response to comments by Franklin T. Adams-Watters - Wolfdieter Lang, Apr 02 2011.