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 A129529 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} that have k inversions (n >= 0, k >= 0). 1
 1, 3, 6, 3, 10, 8, 8, 1, 15, 15, 21, 18, 9, 3, 21, 24, 39, 45, 48, 30, 24, 9, 3, 28, 35, 62, 82, 107, 108, 101, 81, 62, 37, 17, 8, 1, 36, 48, 90, 129, 186, 222, 264, 252, 255, 219, 183, 126, 90, 48, 27, 9, 3, 45, 63, 123, 186, 285, 372, 492, 561, 624, 648, 651, 597, 537, 435 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row n has 1 + floor(n^2/3) terms. Row sums are equal to 3^n = A000244(n). Alternating row sums are 3^(ceiling(n/2)) = A108411(n+1). T(n,0) = (n+1)*(n+2)/2 = A000217(n+1). Sum_{k>=0} k*T(n,k) = 3^(n-1)*n*(n-1)/2 = A129530(n). This sequence is mentioned in the Andrews-Savage-Wilf paper. - Omar E. Pol, Jan 30 2012 REFERENCES M. Bona, Combinatorics of Permutations, Chapman & Hall/CRC, Boca Raton, FL, 2004, pp. 57-61. G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976. LINKS Table of n, a(n) for n=0..66. G. E. Andrews, C. D. Savage and H. S. Wilf, Hypergeometric identities associated with statistics on words Mark A. Shattuck and Carl G. Wagner, Parity Theorems for Statistics on Lattice Paths and Laguerre Configurations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.1. FORMULA Generating polynomial of row n is Sum_{i=0..n} Sum_{j=0..n-i} binomial[n; i,j,n-i-j], where binomial[n;a,b,c] (a+b+c=n) is a q-multinomial coefficient. EXAMPLE T(3,2)=8 because we have 100, 110, 120, 200, 201, 211, 220 and 221. Triangle starts: 1; 3; 6, 3; 10, 8, 8, 1; 15, 15, 21, 18, 9, 3; 21, 24, 39, 45, 48, 30, 24, 9, 3; MAPLE for n from 0 to 40 do br[n]:=sum(q^i, i=0..n-1) od: for n from 0 to 40 do f[n]:=simplify(product(br[j], j=1..n)) od: mbr:=(n, a, b, c)->simplify(f[n]/f[a]/f[b]/f[c]): for n from 0 to 9 do G[n]:=sort(simplify(sum(sum(mbr(n, a, b, n-a-b), b=0..n-a), a=0..n))) od: for n from 0 to 9 do seq(coeff(G[n], q, j), j=0..floor(n^2/3)) od; # yields sequence in triangular form CROSSREFS Cf. A000244, A083906, A108411, A000217, A129530, A129531, A129532. Sequence in context: A263333 A328371 A134440 * A298263 A128503 A210193 Adjacent sequences: A129526 A129527 A129528 * A129530 A129531 A129532 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Apr 22 2007 STATUS approved

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Last modified March 3 17:50 EST 2024. Contains 370512 sequences. (Running on oeis4.)