|
|
A145597
|
|
Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.
|
|
6
|
|
|
1, 3, 3, 6, 15, 6, 10, 45, 45, 10, 15, 105, 189, 105, 15, 21, 210, 588, 588, 210, 21, 28, 378, 1512, 2352, 1512, 378, 28, 36, 630, 3402, 7560, 7560, 3402, 630, 36, 45, 990, 6930, 20790, 29700, 20790, 6930, 990, 45, 55, 1485, 13068, 50820, 98010, 98010
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,2
|
|
COMMENTS
|
T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 2 and which remain in the upper half-plane y >= 0. An example is given in the Example section below.
The current array is the case r = 2 of the generalized Narayana numbers N_r(n,k) := (r + 1)/(n + 1)*binomial(n + 1,k + r)*binomial(n + 1,k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145598 (r = 3) and A145599 (r = 4).
T(n,k) is the number of preimages of the permutation 3214567...(n+3) under West's stack-sorting map that have exactly k+1 descents. - Colin Defant, Sep 15 2018
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = (3/(n+1))*binomial(n+1,k+2)*binomial(n+1,k-1) for n >=2 and 1 <= k <= n-1. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n,2). Row sums A003517.
O.g.f. for column k+2: 3/(k + 1) * y^(k+3)/(1 - y)^(k+5) * Jacobi_P(k,3,1,(1 + y)/(1 - y)).
Identities for row polynomials R_n(x) := sum {k = 1..n-1} T(n,k)*x^k:
x^2*R_(n-1)(x) = 3*(n-1)*(n-2)/((n+1)*(n+2)*(n+3)) * Sum_{k = 0..n} binomial(n + 3,k) * binomial(2n - k,n) * (x - 1)^k;
Sum_{k = 1..n} (-1)^k*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^n = 6/(n+4)*binomial(2n+1,n-2)*x^n = A003517(n)*x^n.
Row generating polynomial R_(n+2)(x) = 3/(n+3)*x*(1-x)^n * Jacobi_P(n,3,3,(1+x)/(1-x)). [Peter Bala, Oct 31 2008]
|
|
EXAMPLE
|
Triangle starts
n\k|..1.....2....3.....4.....5.....6
====================================
.2.|..1
.3.|..3.....3
.4.|..6....15....6
.5.|.10....45...45....10
.6.|.15...105..189...105....15
.7.|.21...210..588...588...210....21
...
Row 4: T(4,1) = 6: the 6 walks of length 4 from (0,0) to (-2,2) are LLUU, LULU, LUUL, ULLU, ULUL and UULL. Changing L to R in these walks gives the 6 walks from (0,0) to (2,2).
T(4,2) = 15: the 15 walks of length 4 from (0,0) to (0,2) are UUUD, UULR, UURL, UUDU,URUL, ULUR, URLU, ULRU, RUUL, LUUR, RLUU, LRUU, RULU, LURU and UDUU.
.
.
*......*......*......y......*......*......*
.
.
*......6......*.....15......*......6......*
.
.
*......*......*......*......*......*......*
.
.
*......*......*......o......*......*......* x axis
.
|
|
MAPLE
|
with(combinat):
T:= (n, k) -> 3/(n+1)*binomial(n+1, k+2)*binomial(n+1, k-1):
for n from 2 to 11 do
seq(T(n, k), k = 1..n-1);
end do;
|
|
MATHEMATICA
|
Table[3/(n+1) Binomial[n+1, k+2]Binomial[n+1, k-1], {n, 2, 20}, {k, n-1}]//Flatten (* Harvey P. Dale, Aug 12 2023 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|