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A348576 Triangle read by rows: T(n,k) is the number of ordered partitions of [n] into k nonempty subsets, in which the first subset has size at least 2, n >= 1 and 1 <= k <= n.

%I A348576 #32 Jan 03 2022 16:29:31
%S A348576 0,1,0,1,3,0,1,10,12,0,1,25,80,60,0,1,56,360,660,360,0,1,119,1372,
%T A348576 4620,5880,2520,0,1,246,4788,26376,58800,57120,20160,0,1,501,15864,
%U A348576 134316,466704,771120,604800,181440,0,1,1012,50880,637020,3238200,8094240,10584000,6955200,1814400,0
%N A348576 Triangle read by rows: T(n,k) is the number of ordered partitions of [n] into k nonempty subsets, in which the first subset has size at least 2, n >= 1 and 1 <= k <= n.
%C A348576 Ordered partitions are also referred to as weak orders.
%H A348576 D. Galvin, G. Wesley and B. Zacovic, <a href="https://arxiv.org/abs/2110.08953">Enumerating threshold graphs and some related graph classes</a>, arXiv:2110.08953 [math.CO], 2021.
%F A348576 T(n,k) = Sum_{j=1..n-1} (n-j)*A173018(n-1, j-1)*binomial(j-1, n-k-1).
%e A348576 For n=3, the ordered partitions of {1,2,3} in which the first block has size at least 2 are 123, 12/3, 13/2 and 23/1, so T(3,1)=1, T(3,2)=3 and T(3,3)=0.
%e A348576 Triangle begins:
%e A348576   0;
%e A348576   1,     0;
%e A348576   1,     3,     0;
%e A348576   1,    10,    12,       0;
%e A348576   1,    25,    80,      60,       0;
%e A348576   1,    56,   360,     660,     360,       0;
%e A348576   1,   119,  1372.    4620,    5880,    2520,        0;
%e A348576   1,   246,  4788,   26376,   58800,   57120,    20160,        0;
%e A348576   1,   501, 15864,  134316,  466704,  771120,   604800,   181440,       0;
%e A348576   1,  1012, 50880,  637020, 3238200, 8094240, 10584000,  6955200, 1814400, 0;
%e A348576   ...
%p A348576 b:= proc(n, t) option remember; expand(`if`(n=0, 1,
%p A348576       add(x*b(n-j, 1)*binomial(n, j), j=t..n)))
%p A348576     end:
%p A348576 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 2)):
%p A348576 seq(T(n), n=1..10);  # _Alois P. Heinz_, Oct 24 2021
%t A348576 eulerian[n_,m_] := eulerian[n,m] =
%t A348576   Sum[((-1)^k)*Binomial[n+1,k]*((m+1-k)^n), {k,0,m+1}] (* eulerian[n, m] is an Eulerian number, counting the permutations of [n] with m descents *);
%t A348576 op2[n_,k_] := op2[n,k] =
%t A348576    Sum[(n-j)*eulerian[n-1,j-1]*Binomial[j-1,n-k-1], {j,1,n-1}] (* op2[n,k] counts ordered partitions on [n] with k parts, with first part having size at least 2 *); Table[op2[n, k],{n,1,12},{k,1,n}]
%o A348576 (PARI) TE(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial( n+1, j)); \\ A008292
%o A348576 T(n,k) = sum(j=1, n-1, (n-j)*TE(n-1,j)*binomial(j-1,n-k-1)); \\ _Michel Marcus_, Oct 24 2021
%Y A348576 Row sums are A053525.
%Y A348576 Cf. A173018, A000247, A001710, A131689.
%K A348576 nonn,tabl
%O A348576 1,5
%A A348576 _David Galvin_, Oct 23 2021