A242628 - OEIS

%I #37 Nov 12 2022 20:07:32

%S 1,2,1,1,3,2,2,2,1,1,1,1,4,3,3,3,2,2,2,2,3,1,2,2,1,2,1,1,1,1,1,1,5,4,

%T 4,4,3,3,3,3,4,2,3,3,2,3,2,2,2,2,2,2,4,1,3,3,1,3,2,1,2,2,2,1,3,1,1,2,

%U 2,1,1,2,1,1,1,1,1,1,1,1,6,5,5,5,4,4,4,4,5,3,4,4,3,4,3,3,3,3,3,3,5,2

%N Irregular table enumerating partitions; n-th row has partitions in previous row with each part incremented, followed by partitions in previous row with an additional part of size 1.

%C This can be calculated using the binary expansion of n; see the PARI program.

%C The n-th row consists of all partitions with hook size (maximum + number of parts - 1) equal to n.

%C The partitions in row n of this sequence are the conjugates of the partitions in row n of A125106 taken in reverse order.

%C Row n is also the reversed partial sums plus one of the n-th composition in standard order (A066099) minus one. - _Gus Wiseman_, Nov 07 2022

%H Alois P. Heinz, <a href="/A242628/b242628.txt">Rows n = 1..12, flattened</a>

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%e The table starts:

%e   1;

%e   2; 1,1;

%e   3; 2,2; 2,1; 1,1,1;

%e   4; 3,3; 3,2; 2,2,2; 3,1 2,2,1 2,1,1 1,1,1,1;

%e   ...

%p b:= proc(n) option remember; `if`(n=1, [[1]],

%p       [map(x-> map(y-> y+1, x), b(n-1))[],

%p        map(x-> [x[], 1], b(n-1))[]])

%p     end:

%p T:= n-> map(x-> x[], b(n))[]:

%p seq(T(n), n=1..7);  # _Alois P. Heinz_, Sep 25 2015

%t T[1] = {{1}};

%t T[n_] := T[n] = Join[T[n-1]+1, Append[#, 1]& /@ T[n-1]];

%t Array[T, 7] // Flatten (* _Jean-François Alcover_, Jan 25 2021 *)

%o (PARI) apart(n) = local(r=[1]); while(n>1,if(n%2==0,for(k=1,#r,r[k]++),r=concat(r,[1]));n\=2);r \\ Generates n-th partition.

%Y Cf. A241596 (another version of this list of partitions), A125106, A240837, A112531, A241597 (compositions).

%Y For other schemes to list integer partitions, please see for example A227739, A112798, A241918, A114994.

%Y First element in each row is A008687.

%Y Last element in each row is A065120.

%Y Heinz numbers of rows are A253565.

%Y Another version is A358134.

%Y Cf. A000120, A029837, A029931, A030303, A066099, A070939, A253566, A355536, A358135, A358136.

%K nonn,tabf

%O 1,2

%A _Franklin T. Adams-Watters_, May 19 2014