login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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
Sequence in context: A263333 A328371 A134440 * A298263 A128503 A210193
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Apr 22 2007
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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