%I A125181
%S A125181 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,
%T A125181 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,
%U A125181 28,84,168,280,56,14,140,140,28,1,1,9,9,36,9,72,84,9,72,36,252,126,36
%N A125181 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength 
               n whose ascent lengths form the k-th 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 k-th partition of the integer n.
%C A125181 Row n has A000041(n) terms (=number of partitions of n). Row sums yield 
               the Catalan numbers (A000108).
%D A125181 R. P. Stanley, Enumerative Combinatorics Vol. 2, Cambridge University 
               Press, Cambridge, 1999; Theorem 5.3.10.
%F A125181 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!/[(n-k+1)!e(1)!e(2)! ... e(j)! ] (Franklin T. Adams-Watters).
%e A125181 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.
%e A125181 Triangle starts:
%e A125181 1;
%e A125181 1,1;
%e A125181 1,3,1;
%e A125181 1,4,2,6,1;
%e A125181 1,5,5,10,10,10,1;
%p A125181 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)+1-q],multiset): k:=nops(p[numbpart(n)+1-q]): 
               s[n,q]:=n!/(n-k+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
%Y A125181 Cf. A000041, A000108.
%Y A125181 Sequence in context: A023577 A134557 A134264 this_sequence A157076 A049999 
               A126015
%Y A125181 Adjacent sequences: A125178 A125179 A125180 this_sequence A125182 A125183 
               A125184
%K A125181 nonn,tabf
%O A125181 1,5
%A A125181 Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 23 2006