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

1, 1, 2, 1, 5, 6, 1, 12, 23, 24, 1, 27, 93, 119, 120, 1, 58, 360, 662, 719, 720, 1, 121, 1312, 3728, 4919, 5039, 5040, 1, 248, 4541, 20160, 35779, 40072, 40319, 40320, 1, 503, 15111, 103345, 259535, 347769, 362377, 362879, 362880

Offset: 1

Author

Peter Luschny, Aug 11 2010

Keywords

• nonn
• tabl

Comments

Partial row sums of A008292 (triangle of Eulerian numbers).

Given by a very similar formula.

Special case: A179457(n,2) = A000325(n) for n > 1 (Grassmannian permutations).

Examples

			1;
1, 2;
1, 5, 6;
1, 12, 23, 24;
1, 27, 93, 119, 120;
1, 58, 360, 662, 719, 720;
1, 121, 1312, 3728, 4919, 5039, 5040;
1, 248, 4541, 20160, 35779, 40072, 40319, 40320;
1, 503, 15111, 103345, 259535, 347769, 362377, 362879, 362880;
		

References

• Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 533.

Links

• Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker and Amanda Welch, Homomesies on permutations -- an analysis of maps and statistics in the FindStat database, arXiv:2206.13409 [math.CO], 2022-2023. (Def. 4.20 and Prop. 4.22.)
• Peter Luschny, Permutation Trees.

Crossrefs

Cf. A008292.

Row sums sequence is 1,3,12,... A001710(n+1) = (n+1)!/2. - Olivier Gérard, Aug 04 2012

Programs

Maple

Eulerian:= (n,k)-> sum((-1)^j*(k-j+1)^n * binomial(n+1,j),j=0..k+1):
s:=(j,n)-> sum(Eulerian(j,k-1), k=1..n):
for i from 1 to 15 do print(seq(s(i,n),n=1..i)) od; # Gary Detlefs, Nov 18 2011

Mathematica

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 *)

Formula

T(n,k) = sum( ((-1)^j*(k-j)^(n+1))*binomial(n+1,j),j=0..k) - Olivier Gérard, Aug 04 2012