A249042 Three-dimensional array of numbers N(r,p,m) read by triangular slices, each slice being read across rows: N(r,p,m) is the number of "r-panes in a (p,m) structure".
1, 1, 1, 2, 1, 3, 4, 1, 6, 6, 1, 6, 7, 7, 24, 18, 1, 14, 36, 24, 1, 10, 11, 25, 70, 46, 15, 100, 180, 96, 1, 30, 150, 240, 120, 1, 15, 16, 65, 165, 101, 90, 455, 690, 326, 31, 360, 1170, 1440, 600, 1, 62, 540, 1560, 1800, 720
Offset: 1
Examples
The initial triangular slices are: 1 - 1 1 2 --- 1 3 4 1 6 6 ----- 1 6 7 7 24 18 1 14 36 24 ---------- 1 10 11 25 70 46 15 100 180 96 1 30 150 240 120 ---------------- 1 15 16 65 165 101 90 455 690 326 31 360 1170 1440 600 1 62 540 1560 1800 720
Links
- I. J. Good, T. N. Tideman, Stirling numbers and a geometric structure from voting theory, Journal of Combinatorial Theory, Series A Volume 23, Issue 1, July 1977, Pages 34-45.
- Warren D. Smith, D-dimensional orderings and Stirling numbers, October 2014.
Crossrefs
Programs
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Mathematica
S1[m_, n_] := Abs[StirlingS1[m, m - n]]; S2[m_, n_] := StirlingS2[m, m - n]; Nr[r_, p_, m_] := S2[m, p - r] Sum[S1[m - p + r, nu], {nu, 0, r}]; Table[Nr[r, p, m], {m, 1, 6}, {p, 0, m - 1}, {r, 0, p}] // Flatten (* Jean-François Alcover, Nov 01 2018 *)
Formula
There is a formula involving Stirling numbers.
Extensions
More terms from Michel Marcus, Aug 28 2015
Comments