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 A286030 Irregular triangle T(n,k) read by rows: Let S be a 3-member set of integers {f,g,h} where f >= g >= h >= 0 and f+g+h = n. Let S(n,k) be an irregular triangle composed of all S listed in reverse lexicographic order by row n. Then T(n,k) = n!*Q/(3*f!*g!*h!), where Q is the number of permutations of S(n,k). (See "Comments" and "Examples" for additional explanation.) 0
 1, 1, 2, 1, 6, 2, 1, 8, 6, 12, 1, 10, 20, 20, 30, 1, 12, 30, 30, 20, 120, 30, 1, 14, 42, 42, 70, 210, 140, 210, 1, 16, 56, 56, 112, 336, 70, 560, 420, 560, 1, 18, 72, 72, 168, 504, 252, 1008, 756, 630, 2520, 560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS See "Example" below for the starting construction of S(n,k) and T(n,k). To understand S(n,k) and Q, consider example S(5,k), i.e., f+g+h = 5, and S(n,k) are listed in reverse lexicographic order. So S(5,k) = {5,0,0}, {4,1,0}, {3,2,0}, {3,1,1}, {2,2,1} k=1..5, respectively. Q is the number of permutations of S(n,k). So Q=3 when S(n,k) = {5,0,0}, {3,1,1} and {2,2,1}; and Q=6 when S(n,k) = {4,1,0} and {3,2,0}. In general, by definition: Q=1 when all members of S(n,k) are equal, Q=3 when S(n,k) contains a pair, and Q=6 when none of the members of S(n,k) is equal. Suppose three equally-matched players are playing a tournament of n games; and for each game there is one winner and two losers. Then S(n,k) is the "overall win record" (where player order does not matter) after n games. Let p be the probability that any S(n,k) occurs after n games. Then p = T(n,k)/3^(n-1). (See also "Example" section.) Generally, when S(n,k) is a z-member set {f,g,h,i..,y}, then Q is the number of permutations of S(n,k), T(n,k) = n!*Q/(z*f!*g!*h!..*y!) and p = T(n,k)/z^(n-1). So when z=2 we get A008314. (Observation prompted by query from Linda Rogers.) For triangle T(n,k): Row sums are 3^(n-1). Row lengths are A001399(n). Final terms in each row are A199127(n). For n >= 3: T(n,2) = 2*n. For n >= 5: T(n,3) = T(n,4) = A002378(n-1) (oblong numbers). For n >= 6: T(n,6) = A007531(n). For n >= 8: T(n,9) = A033487(n-3). LINKS Nicolas Behr, Pawel Sobocinski, Rule Algebras for Adhesive Categories, arXiv:1807.00785 [cs.LO], 2018, also LIPIcs 27th EACSL Annual Conference on Computer Science Logic (CSL 2018), Vol. 119, pp. 11:1-11:21. EXAMPLE Triangle T(n,k) begins: n/k 1    2    3    4    5   6    7     8    9    10    11    12    13    14 1:  1 2:  1,   2 3:  1,   6,   2 4:  1,   8,   6,  12 5:  1,  10,  20,  20,  30 6:  1,  12,  30,  30,  20, 120,  30 7:  1,  14,  42,  42,  70, 210, 140,  210 8:  1,  16,  56,  56, 112, 336,  70,  560,  420, 560 9:  1,  18,  72,  72, 168, 504, 252, 1008,  756, 630, 2520, 560 10: 1,  20,  90,  90, 240, 720, 420, 1680, 1260, 252, 2520, 5040, 3150, 4200 Triangle S(n,k) begins: n/k    1        2        3        4        5        6        7 1:  {1,0,0} 2:  {2,0,0}  {1,1,0} 3:  {3,0,0}  {2,1,0}  {1,1,1} 4:  {4,0,0}  {3,1,0}  {2,2,0}  {2,1,1} 5:  {5,0,0}  {4,1,0}  {3,2,0}  {3,1,1}  {2,2,1} 6:  {6,0,0}  {5,1,0}  {4,2,0}  {4,1,1}  {3,3,0}  {3,2,1}  {2,2,2} T(4,3) = 6 because n=4 and S(4,3) = {2,2,0}; so Q=3 and 3*4!/(3*2!*2!*0!) = 6. Therefore p = 6/27 = 2/9 that the overall win record = {2,2,0} after playing 4 tournament games. CROSSREFS Cf. A001399, A002378, A007531, A008314, A033487, A199127. Cf. A000041 (partition numbers). Sequence in context: A302690 A030304 A248779 * A208905 A208749 A208751 Adjacent sequences:  A286027 A286028 A286029 * A286031 A286032 A286033 KEYWORD nonn,tabf AUTHOR Bob Selcoe, Apr 30 2017 STATUS approved

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Last modified September 20 07:46 EDT 2020. Contains 337264 sequences. (Running on oeis4.)