cp's OEIS Frontend

A112340 Triangle read by rows of numbers b_{n,k}, n>=1, 1<=k<=n such that Product_{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).

%I A112340 #18 Dec 11 2017 03:57:08
%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,
%U A112340 8840,29484,31947,14068,2772,244,9,0,1,1018,27503,131402,199766,124859,36252
%N A112340 Triangle read by rows of numbers b_{n,k}, n>=1, 1<=k<=n such that Product_{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 partitions 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;
%H A112340 N. Bergeron, M. Zabrocki, <a href="https://arxiv.org/abs/math/0509265">The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree</a>, arXiv:math/0509265 [math.CO], 2005.
%H A112340 M. Rosas and B. Sagan, <a href="https://doi.org/10.1090/S0002-9947-04-03623-2">Symmetric Functions in Noncommuting Variables</a>, Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
%H A112340 M. C. Wolf, <a href="http://dx.doi.org/10.1215/S0012-7094-36-00253-3">Symmetric functions for non-commutative elements</a>, 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
%e A112340 Triangle begins:
%e A112340   1;
%e A112340   1,  0;
%e A112340   1,  2,  0;
%e A112340   1,  5,  3,  0;
%e A112340   1, 13, 16,  4, 0;
%e A112340   1, 28, 67, 34, 5, 0;
%e A112340   ...
%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)]);
%t A112340 nmax = 11; b[n_, k_] /; k < 1 || k > n = 0;
%t A112340 coes[m_] := Product[1/(1 - q^n t^k)^b[n, k], {n, 1, m}, {k, 1, m}] - 1 - Sum[ StirlingS2[i, j] q^i t^j, {i, 1, m}, {j, 1, m}] + O[t]^m + O[q]^m // Normal // CoefficientList[#, {t, q}]&;
%t A112340 sol[1] = {b[1, 1] -> 1};
%t A112340 Do[sol[m] = Solve[Thread[(coes[m] /. sol[m - 1]) == 0]], {m, 2, nmax + 1}];
%t A112340 bb = Flatten[Table[sol[m], {m, 1, nmax + 1}]];
%t A112340 Table[b[n, k] /. bb, {n, 1, nmax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 11 2017 *)
%Y A112340 Cf. A008277, A085686, A112339.
%Y A112340 Cf. A087903, A000110.
%K A112340 nonn,tabl
%O A112340 1,5
%A A112340 _Mike Zabrocki_, Sep 05 2005; Aug 06 2006