A100754 Triangle read by rows: T(n,k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 29, 13, 1, 1, 19, 73, 73, 19, 1, 1, 26, 151, 266, 151, 26, 1, 1, 34, 276, 749, 749, 276, 34, 1, 1, 43, 463, 1781, 2762, 1781, 463, 43, 1, 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1

Offset

2,5

Comments

Row n has n-1 terms. Row sums yield the Fine numbers (A000957).

Related to the number of certain sets of non-crossing partitions for the root system A_n (p. 11, Athanasiadis and Savvidou). - Tom Copeland, Oct 19 2014

T(n,k) is the number of permutations pi of [n-1] with k-1 descents such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - Colin Defant, Sep 16 2018

The absolute values of the polynomials at -1 and j (cube root of 1) seem to be given by A126120 and A005043. - F. Chapoton, Nov 16 2021

Links

Alois P. Heinz, Rows n = 2..142, flattened

C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.

Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Formula

T(n,k) = sum(j/(n-j)*C(n-j,k-j)*C(n-j,k), j=0..min(k, n-k)) (n>=2).

G.f.: t*z*r/(1-t*z*r), where r = r(t,z) is the Narayana function defined by r=z*(1+r)*(1+t*r).

From Tom Copeland, Oct 19 2014: (Start)

With offset 0 for A108263 and offset 1 for A132081, row polynomials of this entry P(n,x) = sum(over i, A108263(n,i)*x^i*(1+x)^(n-2*i)) = sum(over i, A132081(n-2,i)*x^i*(1+x)^(n-2*i)).

E.g., P(4,x)= 1*x*(1+x)^(4-2*1) + 2*x^2*(1+x)^(4-2*2) = x + 4 x^2 + x^3.

Equivalently let Q(n,x) be the row polynomials of A108263. Then P(n,x) = (1+x)^n * Q(n,x/(1+x)^2).

E.g., P(4,x)= (1+x)^4 [x/(1+x)^2 + 2 [x/(1+x)^2)^2]].

See Athanasiadis and Savvidou (p. 7). (End)

Example

T(4,2) = 4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD and UUU*DU*DDD, where U=(1,1), D=(1,-1) and * indicates the peaks.
Triangle starts:
  1;
  1,  1;
  1,  4,  1;
  1,  8,  8,  1;
  1, 13, 29, 13, 1;

Maple

T:=(n, k)->sum((j/(n-j))*binomial(n-j, k-j)*binomial(n-j, k), j=0..min(k, n-k)): for n from 2 to 13 do seq(T(n, k), k=1..n-1) od; # yields the sequence in triangular form

Mathematica

T[n_, k_] := Sum[(j/(n-j))*Binomial[n-j, k-j]*Binomial[n-j, k], {j, 0, Min[k, n-k]}]; Table[T[n, k], {n, 2, 13}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

Crossrefs

Cf. A000957.

Sequence in context: A177947 A132789 A319251 * A332307 A296405 A174035

Adjacent sequences: A100751 A100752 A100753 * A100755 A100756 A100757

Keyword

nonn,tabl

Author

Emeric Deutsch, Jan 14 2005

Status

approved