A112340 - OEIS

%I #18 Dec 11 2017 03:57:08

%S 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 123,853,1265,609,107,7,0,1,251,2787,6325,4696,1376,168,8,0,1,506,

%U 8840,29484,31947,14068,2772,244,9,0,1,1018,27503,131402,199766,124859,36252

%N 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 Row sums equal to A085686, second column = A084174 - 1

%C 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 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 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 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 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 Triangle begins:

%e   1;

%e   1,  0;

%e   1,  2,  0;

%e   1,  5,  3,  0;

%e   1, 13, 16,  4, 0;

%e   1, 28, 67, 34, 5, 0;

%e   ...

%p 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 nmax = 11; b[n_, k_] /; k < 1 || k > n = 0;

%t 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 sol[1] = {b[1, 1] -> 1};

%t Do[sol[m] = Solve[Thread[(coes[m] /. sol[m - 1]) == 0]], {m, 2, nmax + 1}];

%t bb = Flatten[Table[sol[m], {m, 1, nmax + 1}]];

%t Table[b[n, k] /. bb, {n, 1, nmax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 11 2017 *)

%Y Cf. A008277, A085686, A112339.

%Y Cf. A087903, A000110.

%K nonn,tabl

%O 1,5

%A _Mike Zabrocki_, Sep 05 2005; Aug 06 2006