cp's OEIS Frontend

A225485 Number of partitions of n that have frequency depth k, an array read by rows.

%I A225485 #22 Sep 27 2023 13:49:45
%S A225485 0,1,1,1,1,1,1,2,1,1,1,1,2,3,1,3,4,3,1,1,4,8,1,1,3,6,9,3,1,2,8,12,7,1,
%T A225485 3,11,17,10,1,1,11,26,17,1,5,19,25,27,1,1,17,44,38,1,3,25,53,52,1,1,3,
%U A225485 29,63,76,4
%N A225485 Number of partitions of n that have frequency depth k, an array read by rows.
%C A225485 Let S = {x(1),...,x(k)} be a multiset whose distinct elements are y(1),...,y(h).  Let f(i) be the frequency of y(i) in S.  Define F(S) = {f(1),..,f(h)}, F(1,S) = F(S), and F(m,S) = F(F(m-1),S) for m>1.  Then lim(F(m,S)) = {1} for every S, so that there is a least positive integer i for which F(i,S) = {1}, which we call the frequency depth of S.
%C A225485 Equivalently, the frequency depth of an integer partition is the number of times one must take the multiset of multiplicities to reach (1). For example, the partition (32211) has frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2) -> (1). - _Gus Wiseman_, Apr 19 2019
%C A225485 From _Clark Kimberling_, Sep 26 2023: (Start)
%C A225485 Below, m^n abbreviates the sum m+...+m of n terms. In the following list, the numbers p_1,...,p_k are distinct, m >= 1, and k >= 1. The forms of the partitions being counted are as follows:
%C A225485    column 1: [n],
%C A225485    column 2: [m^k],
%C A225485    column 3: [p_1^m,...,p_k^m],
%C A225485    column 4: [(p_1^m_1)^m,..., (p_k^m_k)^m], distinct numbers m_i.
%C A225485 Column 3 is of special interest. Assume first that m = 1, so that the form of partition being counted is p = [p_1,...,p_k], with conjugate given by [q_1,...,q_m] where  q_i is the number of parts of p that are >= i. Since the p_i are distinct, the distinct parts of q are the integers 1,2,...,k. For the general case that m >= 1, the distinct parts of q are the integers m,...,km. Let S(n) denote the set of partitions of n counted by column 3. Then if a and b are in the set S*(n) of conjugates of partitions in S(n), and if a > b, then a - b is also in S*(n). Call this the subtraction property. Conversely, if a partition q has the subtraction property, then q must consist of a set of numbers m,..,km for some m. Thus, column 3 counts the partitions of n that have the subtraction property. (End)
%H A225485 Alois P. Heinz, <a href="/A225485/b225485.txt">Rows n = 1..200, flattened</a> (first 40 rows from Clark Kimberling)
%e A225485 The first 9 rows:
%e A225485   n = 1 .... 0
%e A225485   n = 2 .... 1..1
%e A225485   n = 3 .... 1..1..1
%e A225485   n = 4 .... 1..2..1..1
%e A225485   n = 5 .... 1..1..2..3
%e A225485   n = 6 .... 1..3..4..3
%e A225485   n = 7 .... 1..1..4..8..1
%e A225485   n = 8 .... 1..3..6..9..3
%e A225485   n = 9 .... 1..2..8.12..7
%e A225485 For the 7 partitions of 5, successive frequencies are shown here:
%e A225485   5 -> 1 (depth 1)
%e A225485   41 -> 11 -> 2 -> 1 (depth 3)
%e A225485   32 -> 11 -> 2 -> 1 (depth 3)
%e A225485   311 -> 12 -> 11 -> 2 -> 1 (depth 4)
%e A225485   221 -> 12 -> 11 -> 2 -> 1 (depth 4)
%e A225485   2111 -> 13 -> 11 -> 2 -> 1 (depth 4)
%e A225485   11111 -> 5 -> 1 (depth 2)
%e A225485 Summary: 1 partition has depth 1; 1 has depth 2; 2 have 3; and 3 have 4, so that the row for n = 5 is 1..1..2..3 .
%t A225485 c[s_] := c[s] = Select[Table[Count[s, i], {i, 1, Max[s]}], # > 0 &]
%t A225485 f[s_] := f[s] = Drop[FixedPointList[c, s], -2]
%t A225485 t[s_] := t[s] = Length[f[s]]
%t A225485 u[n_] := u[n] = Table[t[Part[IntegerPartitions[n], i]],
%t A225485   {i, 1, Length[IntegerPartitions[n]]}];
%t A225485 Flatten[Table[Count[u[n], k], {n, 2, 25}, {k, 1, Max[u[n]]}]]
%Y A225485 Row sums are A000041.
%Y A225485 Column k = 2 is A032741.
%Y A225485 Column k = 3 is A325245.
%Y A225485 a(n!) = A325272(n).
%Y A225485 Cf. A181819, A182850, A225486, A323014, A323023, A325238, A325239, A325254.
%Y A225485 Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).
%K A225485 nonn,tabf
%O A225485 1,8
%A A225485 _Clark Kimberling_, May 08 2013