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A342587 Triangle, read by rows: T(n,k) is the number of labeled order relations on n nodes in which the longest chain has k nodes (n>=1, 1<=k<=n).

%I A342587 #9 Mar 16 2021 08:29:20
%S A342587 1,1,2,1,12,6,1,86,108,24,1,840,2310,960,120,1,11642,65700,42960,9000,
%T A342587 720,1,227892,2583126,2510760,712320,90720,5040,1,6285806,142259628,
%U A342587 199357704,71310960,11481120,987840,40320,1,243593040,11012710470,21774014640,9501062760,1781015040
%N A342587 Triangle, read by rows: T(n,k) is the number of labeled order relations on n nodes in which the longest chain has k nodes (n>=1, 1<=k<=n).
%C A342587 Corrects Comtet's table for k=4 and 5 in row n=8.
%H A342587 Brendan McKay, <a href="/A342587/b342587.txt">Table of T(n,k) for n = 1..13</a>
%H A342587 L. Comtet, <a href="https://archive.org/details/Comtet_Louis_-_Advanced_Coatorics/page/n35/mode/2up">Advanced Combinatorics</a>, Reidel, 1974, p. 60.
%e A342587 Triangle T(n,k) (with n >= 1 and 1 <= k <= n) begins as follows:
%e A342587   1;
%e A342587   1,      2;
%e A342587   1,     12,       6;
%e A342587   1,     86,     108,      24;
%e A342587   1,    840,    2310,     960,    120;
%e A342587   1,  11642,   65700,   42960,   9000,   720;
%e A342587   1, 227892, 2583126, 2510760, 712320, 90720, 5040;
%e A342587   ...
%Y A342587 Cf. A000142 (diagonal), A001035 (row sums), A055531 (k=2), A055532 (k=3), A055533 (subdiagonal), A055534 (subdiagonal), A081064, A342501 (connected).
%K A342587 nonn,tabl,nice
%O A342587 1,3
%A A342587 _R. J. Mathar_ and _Brendan McKay_, Mar 16 2021