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A124324 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=k<=floor(n/2)).

%I A124324 #60 Jan 23 2023 16:58:30
%S A124324 1,1,1,1,1,4,1,11,3,1,26,25,1,57,130,15,1,120,546,210,1,247,2037,1750,
%T A124324 105,1,502,7071,11368,2205,1,1013,23436,63805,26775,945,1,2036,75328,
%U A124324 325930,247555,27720,1,4083,237127,1561516,1939630,460845,10395,1,8178
%N A124324 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=k<=floor(n/2)).
%C A124324 Row sums are the Bell numbers (A000110).
%C A124324 It appears that the triangles in this sequence and A112493 have identical columns, except for shifts. - _Jörgen Backelin_, Jun 20 2022
%C A124324 Equivalent to Jörgen Backelin's observation, the rows of A112493 may be read off as the diagonals of this entry. - _Tom Copeland_, Sep 24 2022
%H A124324 Alois P. Heinz, <a href="/A124324/b124324.txt">Rows n = 0..200, flattened</a>
%H A124324 Per Alexandersson and Olivia Nabawanda, <a href="https://arxiv.org/abs/2104.04220">Peaks are preserved under run-sorting</a>, arXiv:2104.04220 [math.CO], 2021.
%H A124324 Fufa Beyene and Roberto Mantaci, <a href="https://arxiv.org/abs/2101.07081">Merging-Free Partitions and Run-Sorted Permutations</a>, arXiv:2101.07081 [math.CO], 2021.
%H A124324 Tom Copeland, <a href="https://tcjpn.files.wordpress.com/2022/08/ltx-the-appell-bell-polynomials_-a-trajectory-_-diff-eq-4.pdf">Appell-Bell polynomials: Linking the associated Bell polynomials and the associated reduced inverse refined Eulerian polynomials</a>, 2022.
%H A124324 Tom Copeland, <a href="https://tcjpn.wordpress.com/2022/08/18/the-reduced-inverse-refined-eulerian-polynomials-and-associated-arrays/">The reduced inverse refined Eulerian polynomials and associated arrays</a>, 2022.
%H A124324 Robin Houston, Adam P. Goucher, and Nathaniel Johnston, <a href="https://arxiv.org/abs/2301.06586">A New Formula for the Determinant and Bounds on Its Tensor and Waring Ranks</a>, arXiv:2301.06586 [math.CO], 2023.
%H A124324 O. Nabawanda, F. Rakotondrajao, and A. S. Bamunoba, <a href="https://arxiv.org/abs/2007.03821">Run Distribution Over Flattened Partitions</a>, arXiv:2007.03821 [math.CO], 2020.
%F A124324 E.g.f.: G(t,z) = exp(t*exp(z) - t + (1-t)*z).
%F A124324 T(n,1) = A000295(n) (the Eulerian numbers).
%F A124324 Sum_{k=0..floor(n/2)} k*T(n,k) = A124325(n).
%F A124324 T(2n,n) = A001147(n). - _Alois P. Heinz_, Apr 06 2018
%e A124324 T(4,2) = 3 because we have 12|34, 13|24 and 14|23 (if we take {1,2,3,4} as our 4-set).
%e A124324 Triangle starts:
%e A124324   1;
%e A124324   1;
%e A124324   1,    1;
%e A124324   1,    4;
%e A124324   1,   11,     3;
%e A124324   1,   26,    25;
%e A124324   1,   57,   130,    15;
%e A124324   1,  120,   546,   210;
%e A124324   1,  247,  2037,  1750,   105;
%e A124324   1,  502,  7071, 11368,  2205;
%e A124324   1, 1013, 23436, 63805, 26775, 945;
%e A124324   ...
%p A124324 G:=exp(t*exp(z)-t+(1-t)*z): Gser:=simplify(series(G,z=0,36)): for n from 0 to 33 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 13 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form
%p A124324 # second Maple program:
%p A124324 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A124324       `if`(i>1, x, 1)*binomial(n-1, i-1)*b(n-i), i=1..n)))
%p A124324     end:
%p A124324 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A124324 seq(T(n), n=0..15);  # _Alois P. Heinz_, Mar 08 2015, Jul 15 2017
%t A124324 multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] :=  b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1]*If[i>1, x^j, 1], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, May 22 2015, after _Alois P. Heinz_ *)
%Y A124324 Columns k=0-10 give: A000012, A000295, A112495, A112496, A112497, A290034, A290035, A290036, A290037, A290038, A290039.
%Y A124324 Cf. A000110, A001147, A048993, A124323, A124325, A136394, A112493.
%K A124324 nonn,tabf
%O A124324 0,6
%A A124324 _Emeric Deutsch_, Oct 28 2006