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, 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

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