cp's OEIS Frontend

A138138 A shell model of partitions. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

%I A138138 #13 May 22 2020 13:22:49
%S A138138 1,1,2,1,1,3,1,1,1,2,2,4,1,1,1,1,1,2,3,5,1,1,1,1,1,1,1,2,2,2,3,3,2,4,
%T A138138 6,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,2,5,7,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A138138 1,2,2,2,2,2,3,3,2,2,4,4,4,3,5,2,6,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A138138 A shell model of partitions. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.
%C A138138 The Integrated Diagram of Partitions is a shell model of partitions of a number. Partitions of n contains all partitions of the previous numbers. The number of shells of the partitions of n is equal to n. The number of parts of the last section of the set of partitions of n is A138137(n)=A006128(n)-A006128(n-1) and equal to the number of terms of row n. The number of terms of row n that are equal to 1 is A000041(n-1). The last term of row n is n. The shell model of partitions has several 2D and 3D versions.
%H A138138 Robert Price, <a href="/A138138/b138138.txt">Table of n, a(n) for n = 1..4630, 20 rows.</a>
%e A138138 ........................................
%e A138138 .. Integrated Diagram of Partitions ...
%e A138138 ........... for n = 1 to 9 ............
%e A138138 .......................................
%e A138138 Partition number \ n = 1 2 3 4 5 6 7 8 9
%e A138138 ........................................
%e A138138 .1) A000041(1)= 1 .... 1 1 1 1 1 1 1 1 1
%e A138138 .2) A000041(2)= 2 .... . 2 1 1 1 1 1 1 1
%e A138138 .3) A000041(3)= 3 .... . . 3 1 1 1 1 1 1
%e A138138 .4) .................. . 2 . 2 1 1 1 1 1
%e A138138 .5) A000041(4)= 5 .... . . . 4 1 1 1 1 1
%e A138138 .6) .................. . . 3 . 2 1 1 1 1
%e A138138 .7) A000041(5)= 7 .... . . . . 5 1 1 1 1
%e A138138 .8) .................. . 2 . 2 . 2 1 1 1
%e A138138 .9) .................. . . 3 . . 3 1 1 1
%e A138138 10) .................. . . . 4 . 2 1 1 1
%e A138138 11) A000041(6)=11 .... . . . . . 6 1 1 1
%e A138138 12) .................. . . 3 . 2 . 2 1 1
%e A138138 13) .................. . . . 4 . . 3 1 1
%e A138138 14) .................. . . . . 5 . 2 1 1
%e A138138 15) A000041(7)=15 .... . . . . . . 7 1 1
%e A138138 16) .................. . 2 . 2 . 2 . 2 1
%e A138138 17) .................. . . 3 . . 3 . 2 1
%e A138138 18) .................. . . . 4 . 2 . 2 1
%e A138138 19) .................. . . . 4 . . . 4 1
%e A138138 20) .................. . . . . 5 . . 3 1
%e A138138 21) .................. . . . . . 6 . 2 1
%e A138138 22) A000041(8)=22 .... . . . . . . . 8 1
%e A138138 23) .................. . . 3 . 2 . 2 . 2
%e A138138 24) .................. . . 3 . . 3 . . 3
%e A138138 25) .................. . . . 4 . . 3 . 2
%e A138138 26) .................. . . . . 5 . 2 . 2
%e A138138 27) .................. . . . . 5 . . . 4
%e A138138 28) .................. . . . . . 6 . . 3
%e A138138 29) .................. . . . . . . 7 . 2
%e A138138 30) A000041(9)=30 .... . . . . . . . . 9
%e A138138 .......................................
%e A138138 Triangle begins:
%e A138138 1
%e A138138 1,2
%e A138138 1,1,3,
%e A138138 1,1,1,2,2,4
%e A138138 1,1,1,1,1,2,3,5
%e A138138 1,1,1,1,1,1,1,2,2,2,3,3,2,4,6
%e A138138 1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,2,5,7
%e A138138 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,2,2,4,4,4,3,5,2,6,8
%e A138138 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,2,3,4,2,2,5,4,5,3,6,2,7,9
%t A138138 Table[ConstantArray[{1}, PartitionsP[n - 1]] ~Join~ Reverse@Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], {n, 8}] // Flatten (* _Robert Price_, May 22 2020 *)
%Y A138138 Cf. A000041, A006128, A138137. See A135010 for another version.
%K A138138 nonn,tabf,less
%O A138138 1,3
%A A138138 _Omar E. Pol_, Mar 16 2008, Mar 25 2008