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Search: id:A129529
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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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+0
2
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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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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^(ceil(n/2))=A108411(n+1). T(n,0)=(n+1)(n+2)/2=A000217(n+1). Sum(k*T(n,k),k>=0)=3^(n-1)*n(n-1)/2=A129530(n).
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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.
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.
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FORMULA
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Generating polynomial of row n is Sum(Sum (binom[n; i,j,n-i-j],j=0..n-i),i=0..n), where binom[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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Cf. A000244, A083906, A108411, A000217, A129530, A129531, A129532.
Sequence in context: A019918 A055373 A134440 this_sequence A128503 A120906 A085709
Adjacent sequences: A129526 A129527 A129528 this_sequence A129530 A129531 A129532
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 22 2007
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