A353131 Triangle read by rows of partial Bell polynomials B_{n,k} (x_1,...,x_{n-k+1}) evaluated at 2, 2, 12, 72, ..., (n-k)(n-k+1)!, n>=1, 1<=k<=n.

2, 2, 4, 12, 12, 8, 72, 108, 48, 16, 480, 960, 600, 160, 32, 3600, 9360, 7320, 2640, 480, 64, 30240, 100800, 95760, 42000, 10080, 1344, 128, 282240, 1189440, 1350720, 700560, 201600, 34944, 3584, 256, 2903040, 15240960, 20442240, 12337920, 4142880, 854784, 112896, 9216, 512

Offset

1,1

Links

Jordan Weaver, Rows 1 to 40 of triangle, flattened

E. T. Bell, Partition polynomials, Ann. Math., 29 (1927-1928), 38-46.

E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.

Sara C. Billey and Jordan E. Weaver, Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs, arXiv:2207.06508 [math.CO], 2022.

A. Knutson, T. Lam and D. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752.

A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

Formula

T(n,k) = A353132(n,k)*(n-k+1)!.

Sum_{k=1..n} T(n,k)/(n-k+1)! = A349458(n).

Example

For n=4,k=2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3)=4x_1x_3+3x_2^2, so a(4,2)=B_{4,2}(2,2,12)=4*2*12+3*2^2=108.
Triangle starts:
[1]      2;
[2]      2,       4;
[3]     12,      12,       8;
[4]     72,     108,      48,     16;
[5]    480,     960,     600,    160,     32;
[6]   3600,    9360,    7320,   2640,    480,    64;
[7]  30240,  100800,   95760,  42000,  10080,  1344,  128;
[8] 282240, 1189440, 1350720, 700560, 201600, 34944, 3584, 256.

Crossrefs

Cf. A000079, A353132, A349413, A268441, A178867.

Sequence in context: A059343 A285944 A112473 * A134435 A136718 A362709

Adjacent sequences: A353128 A353129 A353130 * A353132 A353133 A353134

Keyword

nonn,tabl

Author

Jordan Weaver, Apr 24 2022

Status

approved