A238326 - OEIS

%I #4 Feb 26 2014 18:38:57

%S 1,2,3,4,1,5,2,6,3,2,7,4,4,8,5,6,3,9,6,8,6,1,10,7,10,9,4,2,11,8,12,12,

%T 8,3,2,12,9,14,15,12,5,4,4,2,13,10,16,18,16,10,5,6,3,4,14,11,18,21,20,

%U 15,6,6,8,6,6,4,15,12,20,24,24,20,7,12,10,9,8

%N Array: row n gives the number of occurrences of each possible diagonal partition of n, arranged in reverse Mathematica order.

%C Suppose that p is a partition of n, let F(p) be its Ferrers matrix, as defined at A237981, and let mXm be the size of F(p).  The numbers of 1s in each of the 2m-1 diagonals of F(p) form a partition of n.  Any partition which is associated with a partition of n in this manner is introduced here as a diagonal partition of n.  A000041(n) = sum of the numbers in row n; A003114(n) = number of terms in row n.  Every diagonal partition is an antidiagonal partition, as in A238325 (but not conversely).

%H Clark Kimberling, <a href="/A238326/b238326.txt">Table of n, a(n) for n = 1..1000</a>

%e The Mathematica ordering of the 3 antidiagonal partitions of 6 follows:  2211, 21111, 111111.)  Frequencies of these among the 11 partitions of 6 are given in reverse Mathematica ordering as follows:  111111 occurs 6 times, 21111 occurs 3 times, and 2211 occurs 2 times, so that row 6 of the array is 6 3 2.

%e ...

%e First 9 rows:

%e 1

%e 2

%e 3

%e 4 1

%e 5 2

%e 6 3 2

%e 7 4 4

%e 8 5 6 3

%e 9 6 8 6 1

%t z = 20; ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; diagPartSE[list_] := Module[{m = ferrersMatrix[list]}, Map[Diagonal[m, #] &, Range[-#, #] &[Length[m] - 1]]]; Tally[Map[  DeleteCases[Reverse[Sort[Map[Count[#, 1] &, diagPartSE[#]]]], 0] &, IntegerPartitions[z]]]; a1[n_] := Last[Transpose[Tally[Map[DeleteCases[Reverse[Sort[Map[Count[#, 1] &, diagPartSE[#]]]], 0] &, IntegerPartitions[n]]]]]; t = Table[a1[n], {n, 1, z}]; u = Flatten[t]

%t Map[Last[Transpose[Tally[Map[DeleteCases[Reverse[Sort[Map[Count[#, 1] &, diagPartSE[#]]]], 0] &, IntegerPartitions[#]]]]] &, Range[z]] // TableForm

%t (* _Peter J. C. Moses_, Feb 25 2014 *)

%Y Cf. A238325, A003114, A000041.

%K nonn,tabf,easy

%O 1,2

%A _Clark Kimberling_ and _Peter J. C. Moses_, Feb 25 2014