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A129175 Triangle read by rows: the q-analog of the Catalan numbers. 6
1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 3, 5, 5, 7, 6, 9, 7, 9, 8, 9, 7, 9, 6, 7, 5, 5, 3, 4, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 13, 12, 16, 16, 19, 18, 22, 20, 23, 21, 23 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,17

COMMENTS

Previous name: T(n,k) is the number of Dyck words of length 2n having major index k (n>=0, k>=0). 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. The major index of a Dyck word is the sum of the positions of those 1's that are followed by a 0.

Representing a Dyck word p of length 2n as a Dyck path p', the major index of p is equal to the sum of the abscissae of the valleys of p'.

Row n has 1+n*(n-1) terms.

Row sums are the Catalan numbers (A000108).

T(n,k) = T(n,n^2-n-k) (i.e. rows are palindromic).

Alternating row sums are the central binomial coefficients binom(n,floor(n/2)) = A001405(n).

Sum(k*T(n,k),k=0..n*(n-1)) = A002740(n+1).

T(n,k) = A129174(n,n+k) (i.e. except for the initial 0's, rows of A129174 and A129175 are the same).

REFERENCES

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see p. 236, Exercise 6.34 b. [From Emeric Deutsch, Nov 05 2008]

LINKS

Alois P. Heinz, Rows n = 0..32, flattened

FindStat - Combinatorial Statistic Finder, The major index of a Dyck path.

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.

FORMULA

The generating polynomial for row n is P[n](t) = binom[2n,n]/[n+1], where [n+1]=1+t+t^2+...+t^n and binom[2n,n] is a Gaussian polynomial.

EXAMPLE

T(4,8)=2 because we have 01001101 (with 10's starting at positions 2 and 6) and 00101011 (with 10's starting at positions 3 and 5).

Triangle starts:

1;

1;

1,0,1;

1,0,1,1,1,0,1;

1,0,1,1,2,1,2,1,2,1,1,0,1;

1,0,1,1,2,2,3,2,4,3,4,3,4,2,3,2,2,1,1,0,1;

MAPLE

br:=n->sum(q^i, i=0..n-1): f:=n->product(br(j), j=1..n): cbr:=(n, k)->f(n)/f(k)/f(n-k): P:=n->sort(expand(simplify(cbr(2*n, n)/br(n+1)))): for n from 0 to 7 do seq(coeff(P(n), q, k), k=0..n*(n-1)) 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, true)+b(x-1, y+1, false)*`if`(t, z^x, 1))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, false)):

seq(T(n), n=0..8);  # Alois P. Heinz, Sep 15 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, True] + b[x-1, y+1, False]*If[t, z^x, 1]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, False]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-Fran├žois Alcover, May 26 2015, after Alois P. Heinz *)

p[n_] := QBinomial[2n, n, q]/QBinomial[n+1, 1, q]; Table[CoefficientList[p[n] // FunctionExpand, q], {n, 0, 9}] // Flatten (* Peter Luschny, Jul 22 2016 *)

PROG

(Sage)

from sage.combinat.q_analogues import q_catalan_number

def T(n): return q_catalan_number(n).subs(q=SR.var('x')).list()

for n in (0..6): print T(n) # Peter Luschny, Jul 19 2016

CROSSREFS

Cf. A000108, A001405, A002740, A129174.

Sequence in context: A212213 A214339 A129174 * A063053 A063050 A189024

Adjacent sequences:  A129172 A129173 A129174 * A129176 A129177 A129178

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Apr 20 2007

EXTENSIONS

New name from Peter Luschny, Jul 24 2016

STATUS

approved

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Last modified June 24 10:00 EDT 2019. Contains 324323 sequences. (Running on oeis4.)