|
Search: id:A055105
|
|
|
| A055105 |
|
Triangle read by rows: T(n,k) = number of noncommutative symmetric polynomials of degree n that have exactly k different variables appearing in each monomial and which generate the algebra of all noncommutative symmetric polynomials (n >= 1, 1 <= k <= n). |
|
+0
16
|
|
| 1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 12, 8, 1, 0, 1, 33, 44, 13, 1, 0, 1, 88, 208, 109, 19, 1, 0, 1, 232, 910, 753, 223, 26, 1, 0, 1, 609, 3809, 4674, 2091, 405, 34, 1, 0, 1, 1596, 15521, 27161, 17220, 4926, 677, 43, 1, 0, 1, 4180, 62185, 151134, 130480, 51702, 10342
(list; table; graph; listen)
|
|
|
OFFSET
|
1,9
|
|
|
COMMENT
|
Also the number of irreducible set partitions of size n and length k. A set partition of [n] of length k is a set of sets A = {A_1,A_2,...,A_k} where A_i are nonempty and their union is {1..n}. A set partition A of n+m is reducible if there are B and C which are set partitions of [n] and [m] (resp.), n,m >=1, such that A = { B_1 U (C_1+n),B_2 U (C_2+n),...,B_k U (C_k+n) } (here C_i+n means add n to every entry in C_i). A set partition is irreducible if it is not reducible. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Feb 04 2005
|
|
REFERENCES
|
N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, http://arXiv.org/abs/math.CO/0502082
M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables. Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
|
|
FORMULA
|
Let B_k(q) = sum(sum(S_{n, i}, i=1..k), n>=0) where S_{n, i} are the Stirling numbers of the second kind. then A_k(q) = 1/B_{k-1}(q)-1/B_k(q) is the generating function for the k^th column of this table (k>=0) A(q, t) = sum(t^k(t-1)/B_k(q), k>=0) = sum(sum(T_{n, k}*q^n*t^k, k=1..n), n>=0) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Feb 04 2005
|
|
EXAMPLE
|
T(1,1)=1 from Sum x_1; T(2,2)=1 from Sum x_1 x_2; T(3,2)=1 from Sum x_1 x_2 x_1; T(3,3)=1 from Sum x_1 x_2 x_3; ...
1; 0,1; 0,1,1; 0,1,4,1; 0,1,12,8,1; ...
T(4,3) = 4 because {1|23|4}, {1|2|34}, {1|24|3}, {13|2|4} are irreducible set partitions of size 4 and length 3 while {12|3|4}={1}*{1|2|3}, {14|2|3}={1|2|3}*{1} are both reducible
|
|
MAPLE
|
Bk:=proc(k, n) local i, j; 1+add(add(stirling2(i, j), j=1..k)*q^i, i=1..n); end: Ak:=proc(k, n); series(1/Bk(k-1, n)-1/Bk(k, n), q, n+1); end: T:=proc(n, k); coeff(Ak(k, n), q, n); end: (Zabrocki)
|
|
CROSSREFS
|
Row sums are A074664. Cf. A055106, A055107.
Cf. A000110, A008277.
Sequence in context: A086329 A085852 A123125 this_sequence A058710 A124539 A096501
Adjacent sequences: A055102 A055103 A055104 this_sequence A055106 A055107 A055108
|
|
KEYWORD
|
nonn,tabl,nice
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com), Jun 14 2000
|
|
EXTENSIONS
|
More terms from Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Feb 04 2005
|
|
|
Search completed in 0.002 seconds
|
