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A238606 Array: t(n,k) = number of partitions p of n such that the principal antidiagonal of the Ferrers matrix of p has k 1's.

%I A238606 #13 Apr 07 2020 22:23:56
%S A238606 1,2,2,1,4,1,4,3,8,2,1,10,3,2,14,7,1,20,5,5,30,5,6,1,36,15,2,3,52,16,
%T A238606 6,3,70,13,15,3,94,22,12,7,122,32,8,13,1,160,45,12,10,4,206,51,26,8,6,
%U A238606 276,49,44,9,7,350,75,30,28,7,448,108,22,38,11,566
%N A238606 Array: t(n,k) = number of partitions p of n such that the principal antidiagonal of the Ferrers matrix of p has k 1's.
%C A238606 "Ferrers matrix" is defined (A237981) as follows:  an m X m matrix (x(i,j)) of 0's and 1's satisfying three properties: (1) x(1,m) = 1 or x(m,1) = 1; (2) x(i,j+1) >= x(i,j) for j=1..m-1 and i = 1..m; and (3) x(i+1,j) >= x(i,j) for i=1..m-1 and j=1..m.  Ferrers matrices arise from Ferrers graphs of partitions, as follows:  suppose that p is a partition of n, and let m = max{greatest part of p, number of parts of p}.  Write the Ferrers graph of p with 1s as nodes, and pad the graph with 0's to form an m X m square matrix, which is the Ferrers matrix of p.
%C A238606 If "antidiagonal" is changed to "diagonal" in the definition of t(n,k), the resulting array is given by A115995.  For both arrays, the sum of terms in row n is A000041(n).
%H A238606 Clark Kimberling and Peter J. C. Moses, <a href="http://faculty.evansville.edu/ck6/GalleryThree/Introduction3.html">Ferrers Matrices and Related Partitions of Integers</a>
%e A238606 First 17 rows:
%e A238606 1
%e A238606 2
%e A238606 2 .... 1
%e A238606 4 .... 1
%e A238606 4 .... 3
%e A238606 8 .... 2 .... 1
%e A238606 10 ... 3 .... 2
%e A238606 14 ... 7 .... 1
%e A238606 20 ... 5 .... 5
%e A238606 30 ... 5 .... 6 ... 1
%e A238606 36 ... 15 ... 2 ... 3
%e A238606 52 ... 16 ... 6 ... 3
%e A238606 70 ... 13 ... 15 .. 3
%e A238606 94 ... 22 ... 12 .. 7
%e A238606 122 ... 32 .. 8 ... 13 .. 1
%e A238606 160 ... 45 .. 12 .. 10 .. 4
%e A238606 Row 5 counts 4 antidiagonals that have exactly one 1 and 3 antidiagonals that have exactly two 1's.  The Ferrers matrix for each of the latter three cases are as shown below.
%e A238606 For the partition 32:
%e A238606 1 1 1
%e A238606 1 1 0
%e A238606 0 0 0  (antidiagonal, from row 1:  1,1,0)
%e A238606 For the partition 311:
%e A238606 1 1 1
%e A238606 1 0 0
%e A238606 1 0 0  (antidiagonal, from row 1:  1,0,1,)
%e A238606 For the partition 221:
%e A238606 1 1 0
%e A238606 1 1 0
%e A238606 1 0 0  (antidiagonal, from row 1:  0,1,1)
%t A238606 z = 30; ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; diagAntidiagDots[list_] := {Total[Diagonal[#]], Total[Diagonal[Reverse[#]]]} &[ferrersMatrix[list]]; u[n_, k_] := Length[Select[   Map[diagAntidiagDots, IntegerPartitions[n]], #[[2]] == k &]]; t[n_] := t[n] = Floor[(-1 + Sqrt[1 + 8 n])/2]; w = Table[u[n, k], {n, 1, z}, {k, 1, t[n]}]; y = Flatten[w] (* A238606 *) (* _Peter J. C. Moses_, Mar 01 2014 *)
%Y A238606 Cf. A115994, A000041.
%K A238606 nonn,tabf,easy
%O A238606 1,2
%A A238606 _Clark Kimberling_, Mar 01 2014