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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}.

%I A053440 #47 Jul 02 2025 16:01:59
%S A053440 1,3,2,7,12,6,15,50,60,24,31,180,390,360,120,63,602,2100,3360,2520,
%T A053440 720,127,1932,10206,25200,31920,20160,5040,255,6050,46620,166824,
%U A053440 317520,332640,181440,40320,511,18660,204630,1020600,2739240,4233600,3780000,1814400,362880
%N 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}.
%C A053440 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
%H A053440 G. C. Greubel, <a href="/A053440/b053440.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A053440 F. Brenti and V. Welker, <a href="http://dx.doi.org/10.1007/s00209-007-0251-z">f-vectors of barycentric subdivisions</a> Math. Z., 259(4), 849-865, 2008.
%H A053440 Wikipedia, <a href="http://en.wikipedia.org/wiki/Barycentric_subdivision">Barycentric subdivision</a>
%F A053440 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)).
%F A053440 E.g.f.: exp(x)*(exp(x)-1)/(1-y*(exp(x)-1)). - _Vladeta Jovovic_, Apr 13 2003
%F A053440 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
%F A053440 T(n,k) = (k+1)!*Stirling2(n+2, k+2). - _G. C. Greubel_, Nov 19 2017
%e A053440 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
%e A053440 Triangle begins:
%e A053440    1
%e A053440    3      2
%e A053440    7     12      6
%e A053440   15     50     60     24
%e A053440   31    180    390    360    120
%p A053440 a := (n, k) -> (k+1)!*Stirling2(n+2, k+2):
%p A053440 seq(print(seq(a(n, k), k = 0..n)), n = 0..10);
%t A053440 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 *)
%t A053440 Table[(k+1)!*StirlingS2[n+2,k+2], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 19 2017 *)
%o A053440 (PARI) for(n=0,10, for(k=0,n, print1((k+1)!*stirling(n+2,k+2,2), ", "))) \\ _G. C. Greubel_, Nov 19 2017
%Y A053440 Other versions are A028246, A142071.
%Y A053440 Columns k=0..1 are A000225(n+1), A028243(n+2).
%Y A053440 Cf. A000142 (main diagonal), A002050 (row sums), A019538.
%K A053440 nonn,easy,tabl,nice
%O A053440 0,2
%A A053440 _Rob Arthan_, Jan 12 2000
%E A053440 More terms from _James Sellers_, Jan 14 2000