%I A112340
%S A112340 1,1,0,1,2,0,1,5,3,0,1,13,16,4,0,1,28,67,34,5,0,1,60,249,229,65,6,0,1,
%T A112340 123,853,1265,609,107,7,0,1,251,2787,6325,4696,1376,168,8,0,1,506,8840,
%U A112340 29484,31947,14068,2772,244,9,0,1,1018,27503,131402,199766,124859,36252
%N A112340 Triangle read by rows of numbers b_{n,k}, n>=1, 1<=k<=n such that prod_{n,
               k} 1/(1-q^n t^k)^{b_{n,k}} = 1+ sum_{i,j>=1} S_{i,j} q^i t^j where 
               S_{i,j} are entries in the table A008277 (the inverse Euler transformation 
               of the table of Stirling numbers of the second kind).
%C A112340 Row sums equal to A085686 second column = A084174 - 1
%C A112340 The number of set partitions of size n length k which are 'Lyndon,' that 
               is, since all set partititions are isomorphic to sequences of atomic 
               set partitions (A087903), those which are smallest of all rotations 
               of these sequences in lex order (with respect to some ordering on 
               the atomic set partitions) are Lyndon. 1; 1, 0; 1, 2, 0; 1, 5, 3, 
               0; 1, 13, 16, 4, 0;
%D A112340 N. Bergeron, M. Zabrocki, The Hopf algebras of symmetric functions and 
               quasisymmetric functions in non-commutative variables are free and 
               cofree, math.CO/0509265
%D A112340 M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables. 
               Transactions of the American Mathematical Society, 358 (2006), no. 
               1, 215-232.
%D A112340 M. C. Wolf, Symmetric functions for non-commutative elements, Duke Math. 
               J., 2 (1936), 626-637.
%e A112340 There are 6 set partitions of size 4 and length 3, {12|3|4}, {13|2|4}, 
               {14|2|3}, {1|23|4}, {1|24|3}, {1|2|34} and the sequences the correspond 
               to are ({12},{1},{1}), ({13|2}, {1}), ({14|2|3}), ({1},{12},{1}), 
               ({1},{13|2}), ({1},{1},{12}). Now there are three {({12},{1},{1}), 
               ({1},{12},{1}), ({1},{1},{12})} that are rotations of each other 
               and ({1}, {1}, {12}) is the smallest of these, {({13|2}, {1}), ({1},
               {13|2})} are rotations of each other and ({1},{13|2}) is the smallest 
               and ({14|2|3}) is atomic and all atomic s.p. are Lyndon. Hence {1|2|34}, 
               {1|24|3}, {14|2|3} are Lyndon and a(4,3) = 3
%p A112340 EULERitable:=proc(tbl) local ser,out,i,j,tmp; ser:=1+add(add(q^i*t^j*tbl[i][j], 
               j=1..nops(tbl[i])), i=1..nops(tbl)); out:=[]; for i from 1 to nops(tbl) 
               do tmp:=coeff(ser,q,i); ser:=expand(ser*mul(add((-q^i*t^j)^k*choose(abs(coeff(tmp,
               t,j)),k),k=0..nops(tbl)/i), j = 1..degree(tmp,t))); ser:=subs({seq(q^j=0,
               j=nops(tbl)+1..degree(ser,q))},ser); out:=[op(out),[seq(abs(coeff(tmp,
               t,j)), j=1..degree(tmp,t))]]; end do; out; end: EULERitable([seq([seq(combinat[stirling2](n,
               k),k=1..n)],n=1..10)]);
%Y A112340 Cf. A008277, A085686, A112339.
%Y A112340 Cf. A087903, A000110.
%Y A112340 Sequence in context: A092583 A079134 A163940 this_sequence A037186 A004483 
               A085650
%Y A112340 Adjacent sequences: A112337 A112338 A112339 this_sequence A112341 A112342 
               A112343
%K A112340 nonn,tabl
%O A112340 1,5
%A A112340 Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Sep 05 2005; Aug 06 2006