A053440 Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, ..., n+1}.

1, 3, 2, 7, 12, 6, 15, 50, 60, 24, 31, 180, 390, 360, 120, 63, 602, 2100, 3360, 2520, 720, 127, 1932, 10206, 25200, 31920, 20160, 5040, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 362880

Offset: 0

Author

Rob Arthan, Jan 12 2000

Keywords

• nonn
• easy
• tabl
• nice

Comments

T(n,k) is the number of length k+1 sequences of nonempty mutually disjoint subsets of {1,2,...,n+1}. The e.g.f. for the column corresponding to k is exp(x)*(exp(x)-1)^(k+1). - Geoffrey Critzer, Dec 20 2011

Examples

			T(2,1) = 12 because there are 12 such length 2 sequences of subsets of {1,2,3}: ({1},{2}), ({1},{3}), ({2},{3}), ({1},{2,3}), ({2},{1,3}), ({3},{1,2}) with two orderings for each. - _Geoffrey Critzer_, Dec 20 2011
Triangle begins:
   1
   3      2
   7     12      6
  15     50     60     24
  31    180    390    360    120
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• F. Brenti and V. Welker, f-vectors of barycentric subdivisions Math. Z., 259(4), 849-865, 2008.
• Wikipedia, Barycentric subdivision

Crossrefs

Other versions are A028246, A142071.

Columns k=0..1 are A000225(n+1), A028243(n+2).

Cf. A000142 (main diagonal), A002050 (row sums), A019538.

Programs

Maple

a := (n, k) -> (k+1)!*Stirling2(n+2, k+2):
seq(print(seq(a(n, k), k = 0..n)), n = 0..10);

Mathematica

nn = 5; a = Exp[ x] - 1 ; f[list_] := Select[list, # > 0 &];Map[f, Transpose[Table[Drop[Range[0, nn]!CoefficientList[Series[a^k  Exp[x], {x, 0, nn}],x], 1], {k, 1, 5}]]] // Grid (* Geoffrey Critzer, Dec 20 2011 *)
Table[(k+1)!*StirlingS2[n+2,k+2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2017 *)

PARI

for(n=0,10, for(k=0,n, print1((k+1)!*stirling(n+2,k+2,2), ", "))) \\ G. C. Greubel, Nov 19 2017

Formula

T(0,k) = delta(0,k), T(n,k) = delta(0,k) + (k+1)(T(n-1,k-1) + (k+2)T(n-1,k)).

E.g.f.: exp(x)*(exp(x)-1)/(1-y*(exp(x)-1)). - Vladeta Jovovic, Apr 13 2003

T(n,k) = Sum_{i = 0..n} binomial(n+1,i+1)*(k+1)!*Stirling2(i+1,k+1) = (k+1)!*Stirling2(n+2,k+2) (Brenti and Welker). - Peter Bala, Jul 12 2014

T(n,k) = (k+1)!*Stirling2(n+2, k+2). - G. C. Greubel, Nov 19 2017

Extensions

More terms from James Sellers, Jan 14 2000