A353132 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)!, divided by (n-k+1)!, n >= 1, 1 <= k <= n.
2, 1, 4, 2, 6, 8, 3, 18, 24, 16, 4, 40, 100, 80, 32, 5, 78, 305, 440, 240, 64, 6, 140, 798, 1750, 1680, 672, 128, 7, 236, 1876, 5838, 8400, 5824, 1792, 256, 8, 378, 4056, 17136, 34524, 35616, 18816, 4608, 512, 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024
Offset: 1
Examples
For n = 4, k = 2, the partial Bell polynomial is B_{4,2}(x_1,x_2,x_3) = 4*x_1*x_3 + 3*x_2^2, so T(4,2) = B_{4,2}(2,2,12) - (4*2*12 + 3*2^2)/3! = 18.
Triangle begins:
[1] 2;
[2] 1, 4;
[3] 2, 6, 8;
[4] 3, 18, 24, 16;
[5] 4, 40, 100, 80, 32;
[6] 5, 78, 305, 440, 240, 64;
[7] 6, 140, 798, 1750, 1680, 672, 128;
[8] 7, 236, 1876, 5838, 8400, 5824, 1792, 256;
[9] 8, 378, 4056, 17136, 34524, 35616, 18816, 4608, 512;
[10] 9, 580, 8190, 45480, 122682, 175896, 137760, 57600, 11520, 1024.
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.