A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 5, 4, 6, 4, 8, 6, 9, 6, 12, 7, 14, 10, 16, 11, 20, 13, 24, 16, 28, 18, 34, 21, 40, 26, 46, 30, 56, 34, 64, 41, 75, 48, 88, 54, 102, 64, 118, 73, 138, 84, 159, 98, 182, 112, 210, 128, 242, 148, 276, 168, 318

Offset: 0

Author

Gus Wiseman, Feb 23 2022

Keywords

• nonn

Comments

Also odd-length partitions whose run-lengths are all 2's, except for the last, which is 1.

Examples

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3  4  5    6  7    8    9    A    B      C    D      E    F
              221     331  332  441  442  443    552  553    554  663
                                          551         661    662  771
                                          33221       44221       44331
                                                                  55221
		

Crossrefs

The even-length ordered version is A003242, ranked by A351010.

The opposite version is A053251, even-length A351007, any length A351006.

This is the odd-length case of A351005, even-length A035457.

With only equalities we get:

- opposite any length: A351003

- opposite odd-length: A000009 (except at 0)

- opposite even-length: A351012

- any length: A351004

- odd-length: A351594

- even-length: A035363

Without equalities we get:

- opposite any length: A122129 (apparently)

- opposite odd-length: A122130 (apparently)

- opposite even-length: A351008

- any length: A122135 (apparently)

- odd-length: A351595

- even-length: A122134 (apparently)

Cf. A000070, A000984, A027383, A053738, A236559, A236914, A350842, A350844.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[If[EvenQ[i],#[[i]]!=#[[i+1]],#[[i]]==#[[i+1]]],{i,Length[#]-1}]&]],{n,0,30}]