%I A055105
%S A055105 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,
%T A055105 232,910,753,223,26,1,0,1,609,3809,4674,2091,405,34,1,0,1,1596,15521,
%U A055105 27161,17220,4926,677,43,1,0,1,4180,62185,151134,130480,51702,10342
%N 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).
%C A055105 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
%D A055105 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
%D A055105 M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables. 
               Transactions of the American Mathematical Society, 358 (2006), no. 
               1, 215-232.
%D A055105 M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. 
               J. 2 (1936), 626-637.
%F A055105 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
%e A055105 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; ...
%e A055105 1; 0,1; 0,1,1; 0,1,4,1; 0,1,12,8,1; ...
%e A055105 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
%p A055105 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)
%Y A055105 Row sums are A074664. Cf. A055106, A055107.
%Y A055105 Cf. A000110, A008277.
%Y A055105 Sequence in context: A086329 A085852 A123125 this_sequence A058710 A124539 
               A096501
%Y A055105 Adjacent sequences: A055102 A055103 A055104 this_sequence A055106 A055107 
               A055108
%K A055105 nonn,tabl,nice
%O A055105 1,9
%A A055105 N. J. A. Sloane (njas(AT)research.att.com), Jun 14 2000
%E A055105 More terms from Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Feb 04 
               2005