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.

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

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);

R_n(q=2) = A125812; R_n(q=3) = A125813; R_n(q=4) = A125814; R_n(q=5) = A125815.

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

Cf. A000110, A080107, A125811, A125812, A125813, A125814, A125815.

Cf. A264082.

Sequence in context: A207605 A112931 A121685 * A226504 A152195 A133156

Adjacent sequences: A125807 A125808 A125809 * A125811 A125812 A125813

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 10 2006

Status

approved