OFFSET
0,3
COMMENTS
Row n evaluated at sample values of q are as follows:
R_n(q=1) = A000110(n) (Bell numbers);
R_n(q=-1) = A080107(n) (fixed points of permutation of SetPartitions);
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
LINKS
Alois P. Heinz, Rows n = 0..50, flattened
Arvind Ayyer and Naren Sundaravaradan, An area-bounce exchanging bijection on a large subset of Dyck paths, arXiv:2401.14668 [math.CO], 2024. See p. 20.
FORMULA
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)].
Sum_{k>0} k * T(n,k) = A264082(n). - Alois P. Heinz, Apr 03 2016
EXAMPLE
Row g.f.s B_q(n) are polynomials in q generated by:
B_q(n) = Sum_{j=0..n-1} B_q(j) * C_q(n-1,j) for n>0 with B_q(0)=1
where the triangle of q-binomial coefficients C_q(n,k) begins:
1;
1, 1;
1, 1 + q, 1;
1, 1 + q + q^2, 1 + q + q^2, 1;
1, 1 + q + q^2 + q^3, 1 + q + 2*q^2 + q^3 + q^4, 1 + q + q^2 + q^3, 1;
The initial q-Bell coefficients in B_q(n) are:
B_q(0) = 1; B_q(1) = 1; B_q(2) = 2;
B_q(3) = 4 + q;
B_q(4) = 8 + 4*q + 3*q^2;
B_q(5) = 16 + 12*q + 13*q^2 + 8*q^3 + 3*q^4;
B_q(6) = 32 + 32*q + 42*q^2 + 38*q^3 + 33*q^4 + 15*q^5 + 10*q^6 + q^7.
Number of terms in row n is given by A125811, which starts:
1,1,1,2,3,5,8,11,15,20,26,32,39,47,56,66,76,87,99,112,126,141,156,...
Triangle begins:
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, 145, 121, 98, 60, 37, 15, 4;
128, 192, 320, 408, 507, 526, 544, 457, 391, 281, 195, 104, 61, 20, 6;
256, 448, 816, 1160, 1585, 1875, 2189, 2259, 2256, 2066, 1819, 1450, 1133, 777, 506, 300, 158, 65, 25, 4;
512, 1024, 2016, 3136, 4626, 6020, 7642, 8849, 9963, 10423, 10587, 10066, 9355, 8103, 6828, 5380, 4101, 2882, 1964, 1194, 708, 353, 167, 57, 18, 1; ...
MATHEMATICA
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 *)
PROG
(PARI) /* q-Binomial coefficients: */
{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)))}
/* q-Bell numbers = eigensequence of q-binomial triangle: */
{B_q(n) = if(n==0, 1, sum(k=0, n-1, B_q(k)*C_q(n-1, k)))}
/* Coefficients in row n: */
{T(n, k) = polcoeff(B_q(n), k, q)}
/* Print triangle rows: */
for(n=0, 10, for(k=0, #Vec(B_q(n))-1, print1(T(n, k), ", ")); print(" "))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, Dec 10 2006
STATUS
approved