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A274778 Number of proper mergings of an n-antichain and an n-chain. 0
0, 3, 26, 442, 12899, 582381, 37700452, 3315996468, 380835212037, 55380159334315, 9950025870043126, 2165134468142294430, 561245519520167902471, 170913803045738754172185, 60421582956702701927410120, 24543570079301728283314502248, 11353373604627607560431407875081 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is also the number of monotone (n+1)-colorings of a complete bipartite digraph K(n,n), where a monotone (n+1)-coloring is a labeling w of the vertices of K(n,n) with integers in {1,2,...,n+1} such that for every arc (e1, e2) we have w(e1) <= w(e2).

LINKS

Table of n, a(n) for n=0..16.

H. Mühle, Counting Proper Mergings of Chains and Antichains, Discrete Math., Vol. 327(C), 2014, 118-129. Also arXiv:1206.3922 [math.CO], 2012.

FORMULA

a(n) = Sum_{i=1..n+1} ((n+2-i)^n - (n+1-i)^n)*i^n.

EXAMPLE

For n=1, the three proper mergings of a 1-chain {x} and a 1-antichain {y} are x<y, y<x, and x,y.

MAPLE

a := n -> add(((n-i+1)^n-(n-i)^n)*(i+1)^n, i=0..n):

seq(a(n), n=0..16); # Peter Luschny, Nov 11 2016

MATHEMATICA

a[0] = 0; a[n_] := Sum[((n-i+1)^n - (n-i)^n)*(i+1)^n, {i, 0, n}];

Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

PROG

(PARI) a(n) = sum(i=1, n+1, ((n+2-i)^n - (n+1-i)^n)*i^n); \\ Michel Marcus, Jul 14 2018

CROSSREFS

Cf. A085465.

Sequence in context: A182958 A174423 A321183 * A049088 A089041 A059511

Adjacent sequences:  A274775 A274776 A274777 * A274779 A274780 A274781

KEYWORD

nonn

AUTHOR

Henri Mühle, Nov 11 2016

STATUS

approved

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Last modified June 22 16:01 EDT 2021. Contains 345386 sequences. (Running on oeis4.)