|
| |
|
|
A038675
|
|
Triangle read by rows: T(n,k)=A(n,k)*binomial(n+k-1,n), where A(n,k) are the Eulerian numbers (A008292).
|
|
2
|
|
|
|
1, 1, 3, 1, 16, 10, 1, 55, 165, 35, 1, 156, 1386, 1456, 126, 1, 399, 8456, 25368, 11970, 462, 1, 960, 42876, 289920, 393030, 95040, 1716, 1, 2223, 193185, 2577135, 7731405, 5525091, 741741, 6435, 1, 5020, 803440, 19411480, 111675850, 176644468
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
Andrews, Theory of Partitions, (1976), discussion of multisets.
Let a = a_1,a_2,...,a_n be a sequence on the alphabet {1,2,...,n}. Scan a from left to right and create an n-permutation by noting the POSITION of the elements as you come to them in order from least to greatest. See example. T(n,k) is the number of sequences that correspond to such a permutation having exactly n-k descents. [From Geoffrey Critzer, May 19 2010]
|
|
|
REFERENCES
|
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, Reading, Mass., 1994, p. 269 (Worpitzky's identity).
Miklos Bona, Combinatorics of Permutations,Chapman and Hall,2004,page 6. [From Geoffrey Critzer, May 19 2010]
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
1;
1,3;
1,16,10;
1,55,165,35;
1,156,1386,1456,126;
...
If a = 3,1,1,2,4,3 the corresponding 6-permutation is 2,3,4,1,6,5 because the first 1 is in the 2nd position, the second 1 is in the 3rd position,the 2 is in the 4th position, the first 3 is in the first position, the next 3 is in the 6th position and the 4 is in the 5th position of the sequence a. [From Geoffrey Critzer, May 19 2010]
|
|
|
MAPLE
|
A:=(n, k)->sum((-1)^j*(k-j)^n*binomial(n+1, j), j=0..k): T:=(n, k)->A(n, k)*binomial(n+k-1, n): seq(seq(T(n, k), k=1..n), n=1..10);
|
|
|
MATHEMATICA
|
Table[Table[Eulerian[n, k] Binomial[n + k, n], {k, 0, n - 1}], {n, 1, 10}] (* Geoffrey Critzer, Jun 13 2013 *)
|
|
|
CROSSREFS
|
Row sums yield A000312 (Worpitzky's identity).
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
