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A125810 Triangle of q-Bell number coefficients, read by rows that form polynomials in q, giving the eigensequence for the triangle of q-binomial coefficients.

%I A125810 #51 Feb 26 2025 12:05:12
%S A125810 1,1,2,4,1,8,4,3,16,12,13,8,3,32,32,42,38,33,15,10,1,64,80,120,133,
%T A125810 145,121,98,60,37,15,4,128,192,320,408,507,526,544,457,391,281,195,
%U A125810 104,61,20,6,256,448,816,1160,1585,1875,2189,2259,2256,2066,1819,1450,1133,777,506,300,158,65,25,4
%N A125810 Triangle of q-Bell number coefficients, read by rows that form polynomials in q, giving the eigensequence for the triangle of q-binomial coefficients.
%C A125810 Row n evaluated at sample values of q are as follows:
%C A125810 R_n(q=1) = A000110(n) (Bell numbers);
%C A125810 R_n(q=-1) = A080107(n) (fixed points of permutation of SetPartitions);
%C A125810 R_n(q=2) = A125812; R_n(q=3) = A125813; R_n(q=4) = A125814; R_n(q=5) = A125815.
%C A125810 T(n,k) is the number of set partitions of [n] having exactly k inversions. T(5,4)=3: 145|23, 145|2|3, 15|24|3; T(6,6) = 10: 1456|23, 156|234, 156|23|4, 1456|2|3, 146|25|3, 16|245|3, 156|2|34, 16|25|34, 156|2|3|4, 16|25|3|4. - _Alois P. Heinz_, Apr 03 2016
%H A125810 Alois P. Heinz, <a href="/A125810/b125810.txt">Rows n = 0..50, flattened</a>
%H A125810 Arvind Ayyer and Naren Sundaravaradan, <a href="https://arxiv.org/abs/2401.14668">An area-bounce exchanging bijection on a large subset of Dyck paths</a>, arXiv:2401.14668 [math.CO], 2024. See p. 20.
%F A125810 T(n,0) = 2^(n-1) for n>0. G.f. of row n is a polynomial in q, B_q(n), that is generated by the recurrence: B_q(n) = Sum_{j=0..n-1} B_q(j) * C_q(n-1,j) for n>0, with B_q(0)=1. The q-binomial coefficient (also called Gaussian binomial coefficient) is given by: C_q(n,k) = [Product_{i=n-k+1..n} (1-q^i)]/[Product_{j=1..k} (1-q^j)].
%F A125810 Sum_{k>0} k * T(n,k) = A264082(n). - _Alois P. Heinz_, Apr 03 2016
%e A125810 Row g.f.s B_q(n) are polynomials in q generated by:
%e A125810 B_q(n) = Sum_{j=0..n-1} B_q(j) * C_q(n-1,j) for n>0 with B_q(0)=1
%e A125810 where the triangle of q-binomial coefficients C_q(n,k) begins:
%e A125810 1;
%e A125810 1, 1;
%e A125810 1, 1 + q, 1;
%e A125810 1, 1 + q + q^2, 1 + q + q^2, 1;
%e A125810 1, 1 + q + q^2 + q^3, 1 + q + 2*q^2 + q^3 + q^4, 1 + q + q^2 + q^3, 1;
%e A125810 The initial q-Bell coefficients in B_q(n) are:
%e A125810 B_q(0) = 1; B_q(1) = 1; B_q(2) = 2;
%e A125810 B_q(3) = 4 + q;
%e A125810 B_q(4) = 8 + 4*q + 3*q^2;
%e A125810 B_q(5) = 16 + 12*q + 13*q^2 + 8*q^3 + 3*q^4;
%e A125810 B_q(6) = 32 + 32*q + 42*q^2 + 38*q^3 + 33*q^4 + 15*q^5 + 10*q^6 + q^7.
%e A125810 Number of terms in row n is given by A125811, which starts:
%e A125810 1,1,1,2,3,5,8,11,15,20,26,32,39,47,56,66,76,87,99,112,126,141,156,...
%e A125810 Triangle begins:
%e A125810     1;
%e A125810     1;
%e A125810     2;
%e A125810     4,   1;
%e A125810     8,   4,   3;
%e A125810    16,  12,  13,    8,    3;
%e A125810    32,  32,  42,   38,   33,   15,   10,    1;
%e A125810    64,  80, 120,  133,  145,  121,   98,   60,   37,   15,    4;
%e A125810   128, 192, 320,  408,  507,  526,  544,  457,  391,  281,  195,  104,   61,  20, 6;
%e A125810   256, 448, 816, 1160, 1585, 1875, 2189, 2259, 2256, 2066, 1819, 1450, 1133, 777, 506, 300, 158, 65, 25, 4;
%e A125810   ...
%p A125810 b:= proc(o, u, t) option remember; expand(
%p A125810      `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(x^(u+j-1)*
%p A125810         b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o))))
%p A125810     end:
%p A125810 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
%p A125810 seq(T(n), n=0..10);  # _Alois P. Heinz_, Feb 21 2025
%t A125810 QB[n_, q_] := QB[n, q] = Sum[QB[j, q] QBinomial[n-1, j, q], {j, 0, n-1}] // FunctionExpand // Simplify; QB[0, q_]=1; QB[1, q_]=1; Table[ CoefficientList[QB[n, q], q], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Feb 29 2016 *)
%o A125810 (PARI) /* q-Binomial coefficients: */
%o A125810 {C_q(n, k) = if(n<k||k<0, 0, if(n==0||k==0, 1, prod(j=n-k+1, n, 1-q^j)/prod(j=1, k, 1-q^j)))}
%o A125810 /* q-Bell numbers = eigensequence of q-binomial triangle: */
%o A125810 {B_q(n) = if(n==0, 1, sum(k=0, n-1, B_q(k)*C_q(n-1, k)))}
%o A125810 /* Coefficients in row n: */
%o A125810 {T(n, k) = polcoeff(B_q(n), k, q)}
%o A125810 /* Print triangle rows: */
%o A125810 for(n=0, 10, for(k=0, #Vec(B_q(n))-1, print1(T(n, k), ", ")); print(" "))
%Y A125810 Cf. A000110 (row sums), A080107, A125811, A125812, A125813, A125814, A125815.
%Y A125810 Cf. A264082, A381299.
%K A125810 nonn,tabf
%O A125810 0,3
%A A125810 _Paul D. Hanna_, Dec 10 2006