A289871 Irregular triangle read by rows T(n, k) is the number of admissible pinnacle sets with maximum element n and cardinality k.

1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 5, 0, 1, 5, 9, 0, 1, 6, 14, 14, 0, 1, 7, 20, 28, 0, 1, 8, 27, 48, 42, 0, 1, 9, 35, 75, 90, 0, 1, 10, 44, 110, 165, 132, 0, 1, 11, 54, 154, 275, 297, 0, 1, 12, 65, 208, 429, 572, 429, 0, 1, 13, 77, 273, 637, 1001, 1001

Offset

0,10

Comments

See David et al. link for definitions.

Links

Table of n, a(n) for n=0..72.

Robert Davis, Sarah A. Nelson, T. Kyle Petersen, Bridget E. Tenner, The pinnacle set of a permutation, arXiv:1704.05494 [math.CO], 2017.

Formula

T(n,k) = Sum_{n>2k} T(n,k-1) if n>2*k; T(n,k) = 0 if n<=2*k; T(0,0) = 1.

Example

Triangle begins:
1;
0;
0;
0, 1;
0, 1;
0, 1, 2;
0, 1, 3;
0, 1, 4, 5;
0, 1, 5, 9;
...

Mathematica

T[0, 0] = 1; T[_, 0] = 0; T[n_, k_] /; n <= 2k = 0; T[n_, k_] := T[n, k] = Sum[T[j, k-1], {j, 0, n-1}];
Table[T[n, k], {n, 0, 16}, {k, 0, Max[0, (n-1)/2]}] // Flatten (* Jean-François Alcover, Feb 02 2019 *)

Prog

(PARI) T(n, k) = {if ((n==0) && (k==0), return (1)); if (n <= 2*k, return(0)); sum(kk=0, n-1, T(kk, k-1)); }
tabf(nn) = {print(T(0, 0), ", "); for (n=1, nn, for (k=0, round(n-1)\2, print1(T(n, k), ", "); ); print(); ); }

Crossrefs

Cf. A008315, A037952 (row sums).

Sequence in context: A210381 A368591 A029297 * A257563 A373118 A219311

Adjacent sequences: A289868 A289869 A289870 * A289872 A289873 A289874

Keyword

nonn,tabf

Author

Michel Marcus, Jul 14 2017

Status

approved