A320347 Number of partitions of n into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a3, ..., a_{m-1} - a_m are different.

1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 18, 19, 24, 31, 29, 40, 44, 51, 56, 72, 69, 90, 97, 114, 125, 154, 151, 192, 207, 237, 255, 304, 314, 377, 401, 457, 493, 573, 596, 698, 750, 845, 905, 1034, 1104, 1255, 1354, 1507, 1624, 1817, 1955, 2178, 2357, 2605, 2794, 3077, 3380

Offset: 1

Author

Seiichi Manyama, Oct 11 2018

Keywords

• nonn

Comments

In other words, a(n) is the number of strict integer partitions of n with distinct first differences. - Gus Wiseman, Mar 25 2021

Examples

			n = 9
[9]        ooooooooo
------------------------------------
[8, 1]      *******o  a_1 - a_2 = 7.
            oooooooo
------------------------------------
[7, 2]       *****oo  a_1 - a_2 = 5.
             ooooooo
------------------------------------
[6, 3]        ***ooo  a_1 - a_2 = 3.
              oooooo
------------------------------------
[6, 2, 1]         *o  a_2 - a_3 = 1.
              ****oo  a_1 - a_2 = 4.
              oooooo
------------------------------------
[5, 4]         *oooo  a_1 - a_2 = 1.
               ooooo
------------------------------------
a(9) = 6.
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500 (terms 1..100 from Seiichi Manyama)
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

Cf. A000009, A320348.

The equal instead of distinct version is A049980.

The non-strict version is A325325 (ranking: A325368).

The non-strict ordered version is A325545.

The version for first quotients is A342520 (non-strict: A342514).

Cf. A003114, A003242, A005117, A238710, A342530.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[#]&]],{n,0,30}] (* Gus Wiseman, Mar 27 2021 *)