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A129529
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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).
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1
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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