A318806 Triangular array read by rows, where T(n,k) is the number of almost distinct partitions of n in which every part is <= k for 1 <= k <= n.

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 4, 5, 6, 1, 2, 4, 6, 7, 8, 1, 2, 4, 7, 9, 10, 11, 1, 2, 4, 7, 10, 12, 13, 14, 1, 2, 4, 8, 12, 15, 17, 18, 19, 1, 2, 4, 8, 13, 17, 20, 22, 23, 24, 1, 2, 4, 8, 14, 20, 24, 27, 29, 30, 31, 1, 2, 4, 8, 15, 22, 28, 32, 35, 37, 38, 39, 1, 2, 4, 8, 15, 24, 32, 38, 42, 45, 47

Offset: 1

Author

Sara Billey, Sep 04 2018

Keywords

• nonn
• tabl

Comments

An almost distinct partition of n with parts bounded by k is a decreasing sequence of positive integers (a(1), a(2), ..., a(k)) such that n = a(1) + a(2) +...+ a(k), any a(i) > 1 is distinct from all other values, and all a(i) <= k.

Examples

			There are T(5,6) = 7 almost distinct partitions of 6 in which every part is <= 5: [5,1], [4,2], [4,1,1], [3,2,1], [3,1,1,1], [2,1,1,1,1], [1,1,1,1,1,1].
Triangle starts:
1;
1, 2;
1, 2, 3;
1, 2, 3, 4;
1, 2, 4, 5,  6;
1, 2, 4, 6,  7,  8;
1, 2, 4, 7,  9, 10, 11;
1, 2, 4, 7, 10, 12, 13, 14;
1, 2, 4, 8, 12, 15, 17, 18, 19;
1, 2, 4, 8, 13, 17, 20, 22, 23, 24;
...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..3240 (rows 1 <= n <= 80, flattened).
• Sara Billey, Matjaž Konvalinka, and Joshua P. Swanson, Tableaux posets and the fake degrees of coinvariant algebras, arXiv:1809.07386 [math.CO], 2018.

Crossrefs

Cf. A000009, A026820, A008302.

Programs

Mathematica

Array[Table[Count[#, ?(# <= k &)], {k, Max@ #}] &@ DeleteCases[Map[Boole[Flatten@ MapAt[Union, TakeDrop[#, LengthWhile[#, # == 1 &]], -1] == # &@ Reverse@ #] Max@ # &, Reverse@ IntegerPartitions[#]], 0] &, 13] // Flatten (* _Michael De Vlieger, Dec 12 2018 *)