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A263633 Irregular triangle read by rows: row n gives coefficients of n-th ordinary Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into graded lexicographic order.

%I A263633 #90 Jul 31 2024 09:48:09
%S A263633 1,1,1,1,2,1,1,3,2,1,1,1,4,3,3,2,2,1,1,5,4,6,3,6,1,2,2,1,1,1,6,5,10,4,
%T A263633 12,4,3,6,3,3,2,2,2,1,1,7,6,15,5,20,10,4,12,6,12,1,3,6,6,3,3,2,2,2,1,
%U A263633 1,1,8,7,21,6,30,20,5,20,10,30,5,4,12,12,12,12,4,3,6,6,3,3,6,1,2,2,2,2,1
%N A263633 Irregular triangle read by rows: row n gives coefficients of n-th ordinary Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into graded lexicographic order.
%C A263633 "Ordinary" here means in contrast to "exponential", cf. A178867 (see Comtet).
%C A263633 Graded lexicographic order with x[1] > x[2] > ... > x[n] means that monomials are compared first by their total degree, with ties broken by lexicographic order. These monomials correspond to integer partitions.
%C A263633 Row sums are powers of 2. Numbers of terms in rows are partition numbers A000041.
%C A263633 OP_n(-a_1,..,-a_n) = EP_n(a_1,2!*a_2,..,n!*a_n) / n!, where OP_n(a_1,..,a_n) are the partition polynomials of this entry and EP_n, the polynomials of A133314; i.e., the sequences are related as reciprocal o.g.f.s are to reciprocal e.g.f.s. The polynomials play a role in expansion of the iterated Lie derivative (g(x) D_x)^n) formalism for the compositional inversion sketched in A133932. With x[n] = t, the array reduces to the Pascal matrix A007318. - _Tom Copeland_, Sep 19 2016
%C A263633 The signed row partition polynomials can be generated by the Gram determinants of equation 2.23 on page 133 of the Verde-Star paper. E.g., h_3 = -b_1^3 + 2 b_1 b_2 - b_3 corresponds to the third row. The connection to A133314 is obtained by substituting a(k) = k!*b_k = -k!*x[k] and b(k) = k!*h_k in A133314 to compute reciprocals of o.g.f.s rather than e.g.f.s. - _Tom Copeland_, Dec 04 2016
%C A263633 For a relation to lambda operations in K-theory on vector bundles, see p. 218 of Dugger. - _Tom Copeland_, Jul 25 2017
%C A263633 Since E(x) = (1+x_1*x)(1+x_2*x)...(1+x_m*x) is the o.g.f. for the elementary symmetric polynomials e_n(x_1,x_2,...,x_m) and the o.g.f. for the complete symmetric polynomials h_n(x_1,x_2,...,x_m) is H(x) = 1 / E(-x), this entry's partition polynomials with correct signs give either sequence in terms of the other. - _Tom Copeland_, Jan 29 2018
%C A263633 A133314 has an interpretation as weighted surjective mappings. With the connections of this mapping colored and permuted to give mappings distinguished by the order of the colorings (an induced linear ordering by color of the connecting arrows), the signed partition polynomials of this entry, multiplied by n!, are generated. - _Tom Copeland_, Sep 10 2020
%D A263633 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 136, 309.
%H A263633 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015; <a href="https://tcjpn.wordpress.com/2019/09/13/associahedra-noncrossing-partitions-and-an-umbral-algebra-of-power-series/">In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms</a>, 2019; <a href="https://tcjpn.wordpress.com/2020/10/08/appells-and-roses-newton-leibniz-euler-riemann-and-symmetric-polynomials/">Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials</a>, 2020.
%H A263633 D. Dugger, <a href="http://math.uoregon.edu/~ddugger/kgeom.pdf">A Geometric Introduction to K-Theory</a>.
%H A263633 Luis Verde-Star, <a href="https://doi.org/10.1006/aima.1998.1765">Representation of symmetric functions as Gram determinants</a>, Advances in Mathematics, 1 Dec 1998, Vol. 140(1):128-143.
%H A263633 Jin Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Wang/wang53.html">Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
%F A263633 G.f.: 1/(1-Sum_{i >= 1} x_i*t^i) = 1 + Sum_{n >= 1} B_n(x_1, x_2,...)*t^n. [Comtet, p. 136, Eq. [3o'].]
%e A263633 The first few polynomials are:
%e A263633 1, x[1]
%e A263633 2, x[1]^2 + x[2]
%e A263633 3, x[1]^3 + 2*x[1]*x[2] + x[3]
%e A263633 4, x[1]^4 + 3*x[1]^2*x[2] + 2*x[1]*x[3] + x[2]^2 + x[4]
%e A263633 5, x[1]^5 + 4*x[1]^3*x[2] + 3*x[1]^2*x[3] + 3*x[1]*x[2]^2 + 2*x[1]*x[4] + 2*x[2]*x[3] + x[5]
%e A263633 6, x[1]^6 + 5*x[1]^4*x[2] + 4*x[1]^3*x[3] + 6*x[1]^2*x[2]^2 + 3*x[1]^2*x[4] + 6*x[1]*x[2]*x[3] + x[2]^3 + 2*x[1]*x[5] + 2*x[2]*x[4] + x[3]^2 + x[6]
%e A263633 ...
%p A263633 with(Groebner):
%p A263633 A263633_row := proc(n) local EE,t1,t2,Q,F,X,p,L,q,c,r;
%p A263633 EE := add(x[i]*t^i, i=1..2*n);
%p A263633 t1 := 1/(1-EE):
%p A263633 t2 := series(t1, t, 2*n):
%p A263633 Q := k -> expand(coeff(t2, t, k));
%p A263633 X := seq(x[i], i=1..n);
%p A263633 p := Q(n);
%p A263633 L := [];
%p A263633 while p <> 0 do
%p A263633    r := LeadingTerm(p, grlex(X));
%p A263633    c := r[1]; q := r[2];
%p A263633    p := p - c*q;
%p A263633    L := [op(L), c];
%p A263633 od;
%p A263633 L end:
%p A263633 for n from 1 to 8 do A263633_row(n) od; # Program expanded by _Peter Luschny_, Sep 26 2016
%Y A263633 For triangle of coefficients of exponential Bell polynomials see A178867.
%Y A263633 Cf. A000041, A263634.
%Y A263633 Cf. A007318, A133314, A133932.
%K A263633 nonn,tabf
%O A263633 1,5
%A A263633 _N. J. A. Sloane_, Oct 28 2015
%E A263633 More terms and some edits by _Peter Luschny_, Sep 26 2016