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

Author

Michel Marcus, Jul 14 2017

Keywords

• nonn
• tabf

Comments

See David et al. link for definitions.

Examples

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

Links

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

Crossrefs

Cf. A008315, A037952 (row sums).

Programs

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 *)

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(););}

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.