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A179457 Triangle read by rows: number of permutation trees of power n and width <= k.

%I A179457 #20 Mar 12 2025 07:42:05
%S A179457 1,1,2,1,5,6,1,12,23,24,1,27,93,119,120,1,58,360,662,719,720,1,121,
%T A179457 1312,3728,4919,5039,5040,1,248,4541,20160,35779,40072,40319,40320,1,
%U A179457 503,15111,103345,259535,347769,362377,362879,362880
%N A179457 Triangle read by rows: number of permutation trees of power n and width <= k.
%C A179457 Partial row sums of A008292 (triangle of Eulerian numbers).
%C A179457 Given by a very similar formula.
%C A179457 Special case: A179457(n,2) = A000325(n) for n > 1 (Grassmannian permutations).
%D A179457 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 533.
%H A179457 Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker and Amanda Welch, <a href="https://doi.org/10.48550/arxiv.2206.13409">Homomesies on permutations -- an analysis of maps and statistics in the FindStat database</a>, arXiv:2206.13409 [math.CO], 2022-2023. (Def. 4.20 and Prop. 4.22.)
%H A179457 Peter Luschny, <a href="http://www.oeis.org/wiki/User:Peter_Luschny/PermutationTrees">Permutation Trees</a>.
%F A179457 T(n,k) = sum( ((-1)^j*(k-j)^(n+1))*binomial(n+1,j),j=0..k) - _Olivier Gérard_, Aug 04 2012
%e A179457 1;
%e A179457 1, 2;
%e A179457 1, 5, 6;
%e A179457 1, 12, 23, 24;
%e A179457 1, 27, 93, 119, 120;
%e A179457 1, 58, 360, 662, 719, 720;
%e A179457 1, 121, 1312, 3728, 4919, 5039, 5040;
%e A179457 1, 248, 4541, 20160, 35779, 40072, 40319, 40320;
%e A179457 1, 503, 15111, 103345, 259535, 347769, 362377, 362879, 362880;
%p A179457 Eulerian:= (n,k)-> sum((-1)^j*(k-j+1)^n * binomial(n+1,j),j=0..k+1):
%p A179457 s:=(j,n)-> sum(Eulerian(j,k-1), k=1..n):
%p A179457 for i from 1 to 15 do print(seq(s(i,n),n=1..i)) od; # _Gary Detlefs_, Nov 18 2011
%t A179457 Flatten[Table[Table[Sum[(-1)^j (k-j)^(n+1) Binomial[n+1, j], {j, 0, k}], {k, 1, n + 1}], {n, 0, 10}], 1] (* _Olivier Gérard_, Aug 04 2012 *)
%Y A179457 Cf. A008292.
%Y A179457 Row sums sequence is 1,3,12,... A001710(n+1) = (n+1)!/2. - _Olivier Gérard_, Aug 04 2012
%K A179457 nonn,tabl
%O A179457 1,3
%A A179457 _Peter Luschny_, Aug 11 2010