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
Harvey P. Dale, Table of n, a(n) for n = 2..1000
F. Cai, Q.-H. Hou, Y. Sun, and A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018; Table 2.1 for k=2.
Colin Defant, Preimages under the stack-sorting algorithm, arXiv:1511.05681 [math.CO], 2015-2018; Graphs Combin., 33 (2017), 103-122.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
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]
G.f.: x*y*A001263(x,y)^3. - Vladimir Kruchinin, Nov 14 2020
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.
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*......*......*......y......*......*......*
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*......6......*.....15......*......6......*
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*......*......*......*......*......*......*
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*......*......*......o......*......*......* x axis
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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
Peter Bala, Oct 15 2008
STATUS
approved