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

Author

Paul D. Hanna, Dec 10 2006

Keywords

• nonn
• tabf

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

Examples

			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;
  ...
		

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.

Crossrefs

Cf. A000110 (row sums), A080107, A125811, A125812, A125813, A125814, A125815.

Cf. A264082, A381299.

Programs

Maple

b:= proc(o, u, t) option remember; expand(
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(x^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 21 2025

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 *)

PARI

/* q-Binomial coefficients: */
{C_q(n, k) = if(n
				

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