cp's OEIS Frontend

A241596 Partitions listed by alternately incrementing each part and appending a 1.

%I A241596 #39 Sep 25 2015 07:57:19
%S A241596 1,2,11,3,22,21,111,4,33,32,222,31,221,211,1111,5,44,43,333,42,332,
%T A241596 322,2222,41,331,321,2221,311,2211,2111,11111,6,55,54,444,53,443,433,
%U A241596 3333,52,442,432,3332,422,3322,3222,22222,51,441,431,3331,421,3321,3221,22221,411,3311,3211,22211,3111,22111,21111,111111
%N A241596 Partitions listed by alternately incrementing each part and appending a 1.
%C A241596 Start with S_0 = {1}.
%C A241596 Thereafter, S_{n+1} consists of the partitions in S_n with all parts incremented by 1, together with all partitions in S_n with an additional part of 1.
%C A241596 From _Franklin T. Adams-Watters_, May 19 2014:
%C A241596 a(n) can be defined in terms of the binary expansion of n. Start with the partition [1]. Now process the bits of n from right to left, excluding the leading 1. For a zero bit, increase each number in the partition by 1; for a one bit, add a part of size 1. For example, for n=11, binary 1011, we get 1 -> 11 -> 111 -> 222 = a(11).
%C A241596 Row n consists of all partitions with hook size (maximum part + number of parts - 1) equal to n.
%C A241596 This sequence will eventually fail because digits greater than 9 are needed.
%D A241596 Arie Groeneveld, Posting to Sequence Fans List, May 19 2014
%H A241596 Alois P. Heinz, <a href="/A241596/b241596.txt">Rows n = 1..9, flattened</a>
%e A241596 The partitions appear in the following order:
%e A241596 S_0 = 1,
%e A241596 S_1 = 2, 11,
%e A241596 S_2 = 3, 22, 21, 111,
%e A241596 S_3 = 4, 33, 32, 222, 31, 221, 211, 1111,
%e A241596 S_4 = 5, 44, 43, 333, 42, 332, 322, 2222, 41, 331, 321, 2221, 311, 2211, 2111, 11111,
%e A241596 ...
%p A241596 b:= proc(n) option remember; `if`(n=1, [[1]],
%p A241596       [map(x-> map(y-> y+1, x), b(n-1))[],
%p A241596        map(x-> [x[], 1], b(n-1))[]])
%p A241596     end:
%p A241596 T:= n-> map(x-> parse(cat(x[])), b(n))[]:
%p A241596 seq(T(n), n=1..6);
%Y A241596 See A242628 for another version of this list of partitions.
%Y A241596 Cf. A125106, A240837, A112531, A241597 (compositions).
%K A241596 nonn,tabf,base
%O A241596 1,2
%A A241596 _N. J. A. Sloane_, May 19 2014
%E A241596 Typos corrected by _Alois P. Heinz_, Sep 25 2015