cp's OEIS Frontend

A300185 Irregular triangle read by rows: T(n, {j,k}) is the number of partitions of n that have exactly j parts equal to k; 1 <= j <= n, 1 <= k <= n.

%I A300185 #41 Sep 25 2023 14:34:15
%S A300185 1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0,2,1,1,0,
%T A300185 1,2,1,0,0,0,2,0,0,0,0,1,0,0,0,0,1,0,0,0,0,2,2,1,1,0,1,3,1,1,0,0,0,2,
%U A300185 1,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,2,2,1,1,0,1,4,2,1,0,0,0,0,4,1,0,0,0,0,0,3,0
%N A300185 Irregular triangle read by rows: T(n, {j,k}) is the number of partitions of n that have exactly j parts equal to k; 1 <= j <= n, 1 <= k <= n.
%C A300185 Row sums = A027293.
%C A300185 If superfluous zeros are removed from the right side of each row, the row lengths = 1,2,1,3,1,1,4,2,... = A010766.
%C A300185 Sum of each N X N block of rows = 1,2,4,7,12,19,... = A000070.
%C A300185 The sum of the partitions of n that are over-counted in each block of N x N rows = A000070(n) - A000041(n) = A058884(n), n >= 1.
%C A300185 Concatenation of first row from each N X N block = A116598.
%C A300185 As noted by _Joerg Arndt_ in A116598, the first row from each N X N block in reverse converges to A002865.  Two sequences emerge from alternating second rows in reverse:  for 2n, converges to even-indexed terms in A027336, and for 2n+1, converges to odd-indexed terms in A027336.
%C A300185 Counting the rows in each N X N block where columns j=2 > 0 and j=3 through j=n are all zeros produces A008615(n), n > 0.
%H A300185 J. Stauduhar, <a href="/A300185/b300185.txt">Table of n, a(n) for n = 1..10000</a>
%H A300185 Jerome Kelleher and Barry O'Sullivan, <a href="http://arxiv.org/abs/0909.2331">Generating All Partitions: A Comparison Of Two Encodings</a>, arXiv:0909.2331 [cs.DS], 2009-2014.
%H A300185 J. Stauduhar, <a href="/A300185/a300185.py.txt">Original Python program.</a>
%e A300185       \ j  1 2 3 4 5
%e A300185      k
%e A300185 n
%e A300185 1:   1     1
%e A300185 2:   1     0 1
%e A300185      2     1 0
%e A300185 3:   1     1 0 1
%e A300185      2     1 0 0
%e A300185      3     1 0 0
%e A300185 4:   1     1 1 0 1
%e A300185      2     1 1 0 0
%e A300185      3     1 0 0 0
%e A300185      4     1 0 0 0
%e A300185 5:   1     2 1 1 0 1
%e A300185      2     2 1 0 0 0
%e A300185      3     2 0 0 0 0
%e A300185      4     1 0 0 0 0
%e A300185      5     1 0 0 0 0
%e A300185 .
%e A300185 .
%e A300185 .
%t A300185 Array[With[{s = IntegerPartitions[#]}, Table[Count[Map[Count[#, k] &, s], j], {k, #}, {j, #}]] &, 7] // Flatten (* _Michael De Vlieger_, Feb 28 2018 *)
%o A300185 (Python) # See Stauduhar link.
%Y A300185 Cf. A000041, A000070, A008615, A010766, A027293, A027336, A058884, A116598.
%K A300185 nonn,tabf
%O A300185 1,31
%A A300185 _J. Stauduhar_, Feb 27 2018