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A038719 Triangle T(n,k) (0 <= k <= n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.

%I A038719 #75 Apr 29 2022 03:21:38
%S A038719 1,2,1,4,5,2,8,19,18,6,16,65,110,84,24,32,211,570,750,480,120,64,665,
%T A038719 2702,5460,5880,3240,720,128,2059,12138,35406,57120,52080,25200,5040,
%U A038719 256,6305,52670,213444,484344,650160,514080,221760,40320,512,19171
%N A038719 Triangle T(n,k) (0 <= k <= n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.
%C A038719 The relation of this triangle to A143494 given in the Formula section leads to the following combinatorial interpretation: T(n,k) gives the number of partitions of the set {1,2,...,n+2} into k + 2 blocks where 1 and 2 belong to two distinct blocks and the remaining k blocks are labeled from a fixed set of k labels. - _Peter Bala_, Jul 10 2014
%C A038719 Also, the number of distinct k-level fuzzy subsets of a set consisting of n elements ordered by set inclusion. - _Rajesh Kumar Mohapatra_, Mar 16 2020
%H A038719 Alois P. Heinz, <a href="/A038719/b038719.txt">Rows n = 0..140, flattened</a>
%H A038719 Peter Bala, <a href="/A131689/a131689.pdf">Deformations of the Hadamard product of power series</a>
%H A038719 L. Bartlomiejczyk and J. Drewniak, <a href="http://dx.doi.org/10.1007/s00010-004-2737-7">A characterization of sets and operations invariant under bijections</a>, Aequationes Mathematicae 68 (2004), pp. 1-9.
%H A038719 M. Dukes and C. D. White, <a href="http://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016.
%H A038719 Rajesh Kumar Mohapatra and Tzung-Pei Hong, <a href="https://doi.org/10.3390/math10071161">On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences</a>, Mathematics (2022) Vol. 10, No. 7, 1161.
%H A038719 R. B. Nelsen and H. Schmidt, Jr., <a href="http://www.jstor.org/stable/2690450">Chains in power sets</a>, Math. Mag., 64 (1991), 23-31.
%H A038719 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F A038719 T(n, k) = Sum_{j=0..k} (-1)^j*C(k, j)*(k+2-j)^n.
%F A038719 T(n+1, k) = k*T(n, k-1) + (k+2)*T(n, k), T(0,0) = 1, T(0,k) = 0 for k>0.
%F A038719 E.g.f.: exp(2*x)/(1+y*(1-exp(x))). - _Vladeta Jovovic_, Jul 21 2003
%F A038719 A038719 as a lower triangular matrix is the binomial transform of A028246. - _Gary W. Adamson_, May 15 2005
%F A038719 Binomial transform of n-th row = 2^n + 3^n + 4^n + ...; e.g., binomial transform of [8, 19, 18, 6] = 2^3 + 3^3 + 4^3 + 5^3 + ... = 8, 27, 64, 125, ... - _Gary W. Adamson_, May 15 2005
%F A038719 From _Peter Bala_, Jul 09 2014: (Start)
%F A038719 T(n,k) = k!*( Stirling2(n+2,k+2) - Stirling2(n+1,k+2) ).
%F A038719 T(n,k) = k!*A143494(n+2,k+2).
%F A038719 n-th row polynomial = 1/(1 + x)*( sum {k >= 0} (k + 2)^n*(x/(1 + x))^k ). Cf. A028246. (End)
%F A038719 The row polynomials have the form (2 + x) o (2 + x) o ... o (2 + x), where o denotes the black diamond multiplication operator of Dukes and White. See example E12 in the Bala link. - _Peter Bala_, Jan 18 2018
%F A038719 Z(P,m) = Sum_{k=0..n} T(n,k)Binomial(m-2,k) = m^n, the zeta polynomial of the poset B_n. Each length m multichain from 0 to 1 in B_n corresponds to a function from [n] into [m]. - _Geoffrey Critzer_, Dec 25 2020
%F A038719 The entries in row n are the first terms in a table of the successive differences of the sequence [2^n, 3^n, 4^n, ...]. Examples are given below. - _Peter Bala_, Feb 02 2022
%e A038719 Triangle begins
%e A038719    1;
%e A038719    2,   1;
%e A038719    4,   5,   2;
%e A038719    8,  19,  18,   6;
%e A038719   16,  65, 110,  84,  24;
%e A038719   ...
%e A038719 From _Peter Bala_, Feb 02 2022: (Start)
%e A038719 Table of successive differences of k^2 starting at k = 2
%e A038719 4   9   16
%e A038719   5   7
%e A038719     2
%e A038719 gives [4, 5, 2] as row 2 of this triangle.
%e A038719 Table of successive differences of k^3 starting at k = 2
%e A038719 8   27   64   125
%e A038719   19   37   61
%e A038719      18   24
%e A038719         6
%e A038719 gives [8, 19, 8, 6] as row 3 of this triangle. (End)
%p A038719 T:= proc(n, k) option remember;
%p A038719       `if` (n=0, `if`(k=0, 1, 0), k*T(n-1, k-1) +(k+2)*T(n-1, k))
%p A038719     end:
%p A038719 seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Aug 02 2011
%t A038719 t[n_, k_] := Sum[ (-1)^(k-i)*Binomial[k, i]*(2+i)^n, {i, 0, k}]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* _Jean-François Alcover_, after Pari *)
%o A038719 (PARI) T(n,k)=sum(i=0,k,(-1)^(k-i)*binomial(k,i)*(2+i)^n)
%o A038719 (Haskell)
%o A038719 a038719 n k = a038719_tabl !! n !! k
%o A038719 a038719_row n = a038719_tabl !! n
%o A038719 a038719_tabl = iterate f [1] where
%o A038719    f row = zipWith (+) (zipWith (*) [0..] $ [0] ++ row)
%o A038719                        (zipWith (*) [2..] $ row ++ [0])
%o A038719 -- _Reinhard Zumkeller_, Jul 08 2012
%Y A038719 Row sums give A007047. Columns give A000079, A001047, A038721. Next-to-last diagonal gives A038720.
%Y A038719 Diagonal gives A000142. - _Rajesh Kumar Mohapatra_, Mar 16 2020
%Y A038719 Cf. A028246. A019538, A143494.
%K A038719 nonn,easy,nice,tabl
%O A038719 0,2
%A A038719 _N. J. A. Sloane_, May 02 2000
%E A038719 More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000