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A303974
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Regular triangle where T(n,k) is the number of aperiodic multisets of size k that fit within some normal multiset of size n.
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5
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1, 2, 1, 3, 3, 3, 4, 6, 10, 6, 5, 10, 22, 23, 15, 6, 15, 40, 57, 62, 27, 7, 21, 65, 115, 165, 129, 63, 8, 28, 98, 205, 356, 385, 318, 120, 9, 36, 140, 336, 676, 914, 1005, 676, 252, 10, 45, 192, 518, 1176, 1885, 2524, 2334, 1524, 495, 11, 55, 255, 762, 1918, 3528, 5495, 6319, 5607, 3261, 1023
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OFFSET
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1,2
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COMMENTS
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A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.
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LINKS
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FORMULA
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T(n,k) = Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018
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EXAMPLE
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Triangle begins:
1
2 1
3 3 3
4 6 10 6
5 10 22 23 15
6 15 40 57 62 27
7 21 65 115 165 129 63
8 28 98 205 356 385 318 120
9 36 140 336 676 914 1005 676 252
The a(4,3) = 10 multisets: (112), (113), (122), (123), (124), (133), (134), (223), (233), (234).
The a(5,4) = 23 multisets:
(1112), (1222),
(1113), (1123), (1223), (1233), (1333), (2223), (2333),
(1124), (1134), (1224), (1234), (1244), (1334), (1344), (2234), (2334), (2344),
(1235), (1245), (1345), (2345).
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MATHEMATICA
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allnorm[n_Integer]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Length/@GatherBy[Select[Union@@Rest/@Subsets/@allnorm[n], GCD@@Length/@Split[#]===1&], Length], {n, 10}]
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PROG
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(PARI) T(n, k)={sumdiv(k, d, moebius(k/d)*sum(i=1, d, binomial(d-1, i-1)*binomial(n-k+i, i)))} \\ Andrew Howroyd, Sep 18 2018
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CROSSREFS
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Cf. A000740, A000837, A001597, A007716, A007916, A027941, A178472, A210554, A301700, A303431, A303546, A303551, A303945.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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