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 "Mathematica" ordering.
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
Offset: 1
Examples
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 begins:
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
Row 4 counts the following non-crossing set partitions:
{{1234}} {{1}{234}} {{12}{34}} {{1}{2}{34}} {{1}{2}{3}{4}}
{{123}{4}} {{14}{23}} {{1}{23}{4}}
{{124}{3}} {{12}{3}{4}}
{{134}{2}} {{1}{24}{3}}
{{13}{2}{4}}
{{14}{2}{3}}
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
- Germain Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 333-350 (1972).
Crossrefs
Programs
-
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)+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 # second Maple program: b:= proc(n, i, k) `if`(n=0, [k!], `if`(i<1, [], [seq(map(x->x*j!, b(n-i*j, i-1, k-j))[], j=0..n/i)])) end: T:= proc(n) local l, m; l:= b(n, n, n+1); m:=nops(l); seq(n!/l[m-i], i=0..m-1) 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[n-i*j, i-1, k-j]], {j, 0, n/i}]]]; T[n_] := Module[{l, m}, l = b[n, n, n+1]; m = Length[l]; Table[n!/l[[m-i]], {i, 0, m-1}]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *) Table[Binomial[Total[y],Length[y]-1]*(Length[y]-1)!/Product[Count[y,i]!,{i,Max@@y}],{y,Join@@Table[IntegerPartitions[n],{n,1,8}]}] (* Gus Wiseman, Feb 15 2019 *) -
SageMath
def C(p): n = sum(p); l = n - len(p) + 1 def f(x): return factorial(len(list(filter(lambda y: y == x, p)))) return factorial(n) // (factorial(l) * prod(f(x) for x in set(p))) def row(n): return list(C(p) for p in Partitions(n)) for n in range(1, 9): print(row(n)) # Peter Luschny, Jul 14 2022
Formula
Row n has A000041(n) terms (equal to the number of partitions of n).
Row sums yield the Catalan numbers (A000108).
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
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