

A125181


Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n whose ascent lengths form the kth partition of the integer n; the partitions of n are ordered in the way exemplified by [6], [5,1], [4,2], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1], [1,1,1,1,1,1] (the "Mathematica" ordering). Equivalently, T(n,k) is the number of ordered trees with n edges whose node degrees form the kth partition of the integer n.


8



1, 1, 1, 1, 3, 1, 1, 4, 2, 6, 1, 1, 5, 5, 10, 10, 10, 1, 1, 6, 6, 15, 3, 30, 20, 5, 30, 15, 1, 1, 7, 7, 21, 7, 42, 35, 21, 21, 105, 35, 35, 70, 21, 1, 1, 8, 8, 28, 8, 56, 56, 4, 56, 28, 168, 70, 28, 84, 168, 280, 56, 14, 140, 140, 28, 1, 1, 9, 9, 36, 9, 72, 84, 9, 72, 36, 252, 126, 36
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OFFSET

1,5


COMMENTS

Row n has A000041(n) terms (=number of partitions of n). Row sums yield the Catalan numbers (A000108).
For relations to Lagrange inversion through shifted reciprocals of a function, refined Narayana numbers, noncrossing partitions, trees, and other lattice paths, see A134264 and A091867.  Tom Copeland, Nov 01 2014


REFERENCES

R. P. Stanley, Enumerative Combinatorics Vol. 2, Cambridge University Press, Cambridge, 1999; Theorem 5.3.10.


LINKS

Alois P. Heinz, Rows n = 1..26, flattened


FORMULA

Given a partition p=[a(1)^e(1), ..., a(j)^e(j)] into k parts (e(1)+...+e(j)=k), the number of Dyck paths whose ascent lengths yield the partition p is n!/[(nk+1)!e(1)!e(2)! ... e(j)! ].  Franklin T. AdamsWatters


EXAMPLE

Example: T(5,3)=5 because the 3rd partition of 5 is [3,2] and we have (UU)DD(UUU)DDD, (UUU)DDD(UU)DD, (UU)D(UUU)DDDD, (UUU)D(UU)DDDD and (UUU)DD(UU)DDD; here U=(1,1), D=(1,1) and the ascents are shown between parentheses.
Triangle starts:
1;
1, 1;
1, 3, 1;
1, 4, 2, 6, 1;
1, 5, 5, 10, 10, 10, 1;


MAPLE

with(combinat): for n from 1 to 9 do p:=partition(n): for q from 1 to numbpart(n) do m:=convert(p[numbpart(n)+1q], multiset): k:=nops(p[numbpart(n)+1q]): s[n, q]:=n!/(nk+1)!/product(m[j][2]!, j=1..nops(m)) od: od: for n from 1 to 9 do seq(s[n, q], q=1..numbpart(n)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i, k) `if`(n=0, [k!], `if`(i<1, [],
[seq(map(x>x*j!, b(ni*j, i1, kj))[], j=0..n/i)]))
end:
T:= proc(n) local l, m;
l:= b(n, n, n+1); m:=nops(l);
seq(n!/l[mi], i=0..m1)
end:
seq(T(n), n=1..10); # Alois P. Heinz, May 25 2013


MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {k!}, If[i<1, {}, Flatten @ Table[Map[#*j! &, b[ni*j, i1, kj]], {j, 0, n/i}]]]; T[n_] := Module[{l, m}, l = b[n, n, n+1]; m = Length[l]; Table[n!/l[[mi]], {i, 0, m1}]]; Table[T[n], {n, 1, 10}] // Flatten (* JeanFrançois Alcover, May 26 2015, after Alois P. Heinz *)


CROSSREFS

Cf. A000041, A000108.
Cf. A134264, A091867.
Sequence in context: A134557 A219842 A134264 * A157076 A049999 A126015
Adjacent sequences: A125178 A125179 A125180 * A125182 A125183 A125184


KEYWORD

nonn,look,tabf


AUTHOR

Emeric Deutsch, Nov 23 2006


STATUS

approved



