

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.


5



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).


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


CROSSREFS

Cf. A000041, A000108.
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



