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 A187450 Multiset choose 4: number of ways to choose four elements from the multiset representative corresponding to the k-th multiset repetition class defining partition of n in canonical Abramowitz-Stegun order. 1
 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 3, 3, 2, 1, 3, 4, 3, 3, 2, 1, 1, 5, 4, 4, 3, 2, 1, 4, 6, 4, 4, 3, 2, 1, 7, 6, 7, 6, 5, 4, 4, 3, 2, 1, 8, 8, 8, 8, 6, 5, 4, 4, 3, 2, 1, 11, 8, 9, 8, 6, 5, 4, 4, 3, 2, 1, 5, 12, 10, 8, 9, 9, 8, 5, 6, 5, 4, 4, 3, 2, 1, 11, 13, 14, 12, 11, 8, 9, 9, 8, 5, 6, 5, 4, 4, 3, 2, 1, 15, 16, 15, 12, 11, 8, 9, 9, 8, 5, 6, 5, 4, 4, 3, 2, 1, 18, 16, 17, 12, 15, 12, 12, 11, 9, 8, 9, 9, 5, 8, 5, 6, 5, 4, 4, 3, 2, 1, 22, 16, 19, 17, 16, 13, 15, 12, 12, 11, 9, 8, 9, 9, 5, 8, 5, 6, 5, 4, 4, 3, 2, 1, 19, 23, 16, 20, 17, 16, 13, 15, 12, 12, 11, 9, 8, 9, 9, 5, 8, 5, 6, 5, 4, 4, 3, 2, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 COMMENTS The array a(n,k), called "Multiset choose 4", is also denoted by MS(4;n,k). It is the fourth member of an l-family of arrays for multiset choose l, called MS(l;n,k). The row length of this array is A007294=[1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 7,...]. MS(4;n,k) gives for every multiset of the multiset repetition class encoded by the k-th multiset defining partition of n the number of ways to choose four elements from it. The Abramowitz-Stegun (A-St) order for partitions is used (see A036036). For the characteristic array of multiset defining partitions of n in A-St order see A176723. See also the W. Lang links there and in A187447. 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=4 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) for N>=1, and the empty partition for n=N=0.  a(T(N),1)= binomial(N,4). This investigation was stimulated by the Griffiths and Mező paper cited under A176725. LINKS FORMULA a(n,k) gives the number of ways to choose four elements from the multiset representative defined by the k-th multiset repetition class defining partition of n, with k=1,...,A007294(n). a(n,k) = binomial(M(n,k)+3,4) - (m(n,k)[3] + M(n,k)*m(n,k)[2] + binomial(M(n,k)+1,2)*m(n,k)[1] ) + binomial(m(n,k)[1],2), where M(n,k) = MS(1;n,k) := A176725(n,k), and m(n,k)[j] is the j-th member of the multiplicity list m(n,k) of the exponents of the k-th multiset repetition class defining partition of n in A-St order. EXAMPLE n=0..10: 0; 0; 0; 0, 0; 0, 1; 1, 1; 0, 1, 2, 1; 1, 2, 2, 1; 3, 3, 2, 1; 3, 4, 3, 3, 2, 1; 1, 5, 4, 4, 3, 2, 1; ... a(7,2)=MS(4;7,2)=2 because the second multiset repetition class defining partition of n=7 in A-St order is [1^3,2^2] (from the characteristic array A176723) encoding the 5-multiset {1,1,1,2,2}, and there are 2 possibilities to choose 4 elements from this set, namely 1,1,1,2 and 1,1,2,2. a(T(4),1)=a(10,1)=MS(4;10,1) = 1 from the ordinary set {1,2,3,4} (defined by the multiset repetition class defining partition 1,2,3,4). This coincides with binomial(4,4)=1. a(7,3)=MS(4;7,3)=2 from {1,1,1,1,1,2} choose 4 which is 2, namely 1,1,1,1 and 1,1,1,2. The corresponding multiset defining partition has exponent list [5,1] and this is the 16th in the A-St ordered list of such partitions (starting with the empty partition). a(8,3) = 2 = binomial(3+2,4)-(0+0+binomial(2+1,2))+0, because the relevant partition has exponents [6,1] (corresponding to the 7-multiset representative {1,1,1,1,1,1,2}) with MS(1;8,3) = 2, and the multiplicity list of the exponents is m(8,3)=[1,0,0,0,0,1]. Hence m(8,3)[3]=0=m(8,2)[2] and   m(8,2)[1]=1. The two choices are 1,1,1,1 and 1,1,1,2. a(10,3)= 4 = binomial(3+3,4) - (0+0+binomial(3+1,2)*2)+ binomial(2,2) = 15-6*2 +1, corresponding to the 7-multiset {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)[3]=0=m(10,3)[2] and m(10,3)[1]=2. The four choices are 1,1,1,1; 1,1,1,2; 1,1,1,3 and 1,1,2,3. CROSSREFS Cf. A176725: MS(1;n,k), A187445: MS(2;n,k), A187449: MS(3;n,k). Cf. A187447 (all multiset choices; here column l=4). Sequence in context: A261036 A008612 A029320 * A187449 A102541 A243928 Adjacent sequences:  A187447 A187448 A187449 * A187451 A187452 A187453 KEYWORD nonn,easy,tabf AUTHOR Wolfdieter Lang, Apr 04 2011 STATUS approved

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Last modified July 6 11:37 EDT 2022. Contains 355110 sequences. (Running on oeis4.)