A129176 Irregular triangle read by rows: T(n,k) is the number of Dyck words of length 2n having k inversions (n >= 0, k >= 0).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 3, 3, 1, 1, 1, 2, 3, 5, 5, 7, 7, 6, 4, 1, 1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 16, 17, 14, 10, 5, 1, 1, 1, 2, 3, 5, 7, 11, 13, 18, 22, 28, 32, 37, 40, 44, 43, 40, 35, 25, 15, 6, 1, 1, 1, 2, 3, 5, 7, 11, 15, 20, 26, 34, 42, 53, 63, 73, 85, 96, 106, 113, 118, 118, 115, 102, 86, 65, 41, 21, 7, 1

Offset: 0

Author

Emeric Deutsch, Apr 11 2007

Keywords

• nonn
• tabf
• look

Comments

A Dyck word of length 2n is a word of n 0's and n 1's for which no initial segment contains more 1's than 0's.

Representing a Dyck word p of length 2n as a superdiagonal Dyck path p', the number of inversions of p is equal to the area between p' and the path that corresponds to the Dyck word 0^n 1^n.

Row n has 1+n(n-1)/2 terms. Row sums are the Catalan numbers (A000108). Alternating row sums for n>=1 are the Catalan numbers alternated with 0's (A097331). Sum(k*T(n,k),k>=0) = A029760(n-2).

This triangle is A129182 (area under Dyck paths), reflected and compressed (0's removed). Equivalently, A239927 rotated by Pi/2 clockwise and compressed.

This is also the number of Catalan paths of length n and area k. - N. J. A. Sloane, Nov 28 2011

From Alford Arnold, Jan 29 2008:

This triangle gives the partial sums of the following triangle A136624:

1

.1

....2...1

........2...3...3...1

............2...2...6...7...6...4...1

................2...2...4...8..12..15..17..14..10...5...1

etc.

Examples

			T(4,5) = 3 because we have 01010011, 01001101 and 00110101.
Triangle starts:
[0] 1;
[1] 1;
[2] 1, 1;
[3] 1, 1, 2, 1;
[4] 1, 1, 2, 3, 3, 3, 1;
[5] 1, 1, 2, 3, 5, 5, 7,  7,  6,  4,  1;
[6] 1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 16, 17, 14, 10, 5, 1;
...
		

Links

• Alois P. Heinz, Rows n = 0..40, flattened
• Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
• J. Furlinger and J. Hofbauer, q-Catalan numbers, J. Comb. Theory, A, 40, 248-264, 1985.
• M. Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005.

Crossrefs

Cf. A000108, A097331, A029760, A129182, A239927.

Cf. A136624, A136625.

Mirror image of A227543.

Programs

Maple

P[0]:=1: for n from 0 to 8 do
P[n+1]:=sort(expand(sum(t^((i+1)*(n-i))*P[i]*P[n-i],i=0..n))) od:
for n from 1 to 9 do seq(coeff(P[n],t,j),j=0..n*(n-1)/2) od;
# yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, expand(b(x-1, y+1, t)*z^t+b(x-1, y-1, t+1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 10 2014

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, t]*z^t + b[x-1, y-1, t+1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

PARI

P(x, n) =
{
    if ( n<=1, return(1) );
    return( sum( i=0, n-1, P(x, i) * P(x, n-1 -i) * x^((i+1)*(n-1 -i)) ) );
}
for (n=0, 10, print( Vecrev( P(x,n) ) ) ); \\ Joerg Arndt, Jan 23 2024

PARI

\\ faster with memoization:
N=11;
VP=vector(N+1);  VP[1] =VP[2] = 1;  \\ one-based; memoization
P(n) = VP[n+1];
for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) ) );
for (n=0, N, print( Vecrev( P(n) ) ) ); \\ Joerg Arndt, Jan 23 2024

SageMath

from sage.combinat.q_analogues import qt_catalan_number
for n in (0..9): print(qt_catalan_number(n).substitute(q=1).coefficients())
# Peter Luschny, Mar 10 2020

Formula

The row generating polynomials P[n] = P[n](t) satisfy P[0] = 1 and

P[n+1] = Sum_{i=0..n} P[i] P[n-i] t^((i+1)*(n-i)).