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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. 12
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 (list; graph; refs; listen; history; text; internal format)
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, non-crossing partitions, trees, and other lattice paths, see A134264 and A091867. - Tom Copeland, Nov 01 2014

Also the number of non-crossing set partitions whose block sizes are the parts of the n-th integer partition in graded Mathematia ordering. - Gus Wiseman, Feb 15 2019

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).

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!/[(n-k+1)!e(1)!e(2)! ... e(j)! ]. - Franklin T. Adams-Watters

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 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}}

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 *)

CROSSREFS

Other orderings are A134264 and A306438.

Cf. A000041, A000108, A000110, A001263, A016098, A091867, A124794, A134264.

Sequence in context: A290342 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

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Last modified September 30 13:46 EDT 2020. Contains 337439 sequences. (Running on oeis4.)