cp's OEIS Frontend

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (this sequence).

These are partitions containing the sum of some non-singleton submultiset of the parts, a variation of non-binary sum-full partitions where parts cannot be re-used, ranked by A364532. The complement is counted by A237667. The binary version is A237113, or A363225 with re-usable parts. This sequence is weakly increasing. - Gus Wiseman, Aug 12 2023

Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, Chap. 7].

Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes. - Vladeta Jovovic, Dec 27 1999

Also number of different integer triangles with perimeter <= n+3. Also number of different scalene integer triangles with perimeter <= n+9. - Reinhard Zumkeller, May 12 2002

a(n) is the coefficient of q^n in the expansion of (m choose 4)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Also number of partitions of n into parts <= 4. a(n) = A026820(n,4), for n > 3. - Reinhard Zumkeller, Jan 21 2010

Number of different distributions of n+10 identical balls in 4 boxes as x,y,z,p where 0 < x < y < z < p. - Ece Uslu and Esin Becenen, Jan 11 2016

Number of partitions of 5n+8 or 5n+12 into 4 parts (+-) 3 mod 5. a(4) = 5 partitions of 28: [7,7,7,7], [12,7,7,2], [12,12,2,2], [17,7,2,2], [22,2,2,2]. a(3) = 3 partitions of 27: [8,8,8,3], [13,8,3,3], [18,3,3,3]. - Richard Turk, Feb 24 2016

a(n) is the total number of non-isomorphic geodetic graphs of diameter n homeomorphic to a complete graph K4. - Carlos Enrique Frasser, May 24 2018

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.

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

From Clark Kimberling, Sep 26 2023: (Start)

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:

column 1: [n],

column 2: [m^k],

column 3: [p_1^m,...,p_k^m],

column 4: [(p_1^m_1)^m,..., (p_k^m_k)^m], distinct numbers m_i.

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)

At least one part is k and each part is at least k.

From Emeric Deutsch, Feb 19 2006: (Start)

Also number of partitions of n in which the largest part occurs exactly k times. Example: T(6,2)=2 because we have [3,3] and [2,2,1,1].

G.f. of column k is x^k/prod(j>=k, 1-x^j ) (k>=1).

Row sums yield the partition numbers (A000041).

T(n,1) = A000041(n-1) (the partition numbers).

T(n,2) = A002865(n-2) (n>=2).

T(n,3)=A026796(n). T(n,4) = A026797(n). T(n,5) = A026798(n). T(n,6) = A026799(n). T(n,7) = A026800(n). T(n,8) = A026801(n). T(n,9) = A026802(n). T(n,10) = A026803(n).

Sum(k*T(n,k),k=1..n) = A046746(n). (End)

Triangle inverse = A161363. - Gary W. Adamson, Jun 07 2009

T(n,g) is also the number of not necessarily connected 2-regular graphs with girth exactly g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012

From Bob Selcoe, Jul 24 2014 (Start):

Below is a process to generate equations for column k.

Let P be the partition numbers A000041(n-j) and let f(k) denote equations which generate column k.

To find f(k), start with f(1) = P(n-j), j=1. Thus T(n,1) = f(1) = P(n-1). This is the equation for column 1.

To find f(k) k>1, first sum the terms of f(k-1) replacing the value j with j+1, and then subtract the terms of f(k-1) replacing the value j with j+k. So to find f(2) (i.e., the equation for column 2, where k=2), start with f(1) = P(n-1); first replace j with j+1 (yielding P(n-2)), and then replace j with j+2 (yielding P(n-3)). Subtracting the second term from the first, we get: f(2) = P(n-2) - P(n-3).

To find f(3), start with f(2), replace j with j+1 (yielding P(n-3) - P(n-4)) and then replace j with j+3 (yielding P(n-5) - P(n-6)). Subtracting the second group of terms from the first, we get: f(3) = P(n-3) - P(n-4) - P(n-5) + P(n-6). This is the equation for column 3; also the equation for T(n,3) = A026796(n). So for example, T(13,3) = 5 because P(13-3) - P(13-4) - P(13-5) + P(13-6) = 42 - 30 - 22 + 15 = 5.

Continue as above to find f(k) k={4..inf.}. This will generate equations for T(n,4) = A026797(n), T(n,5) = A026798(n), T(n,6) = A026799(n), ad inf.

(End)

Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.

The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).

The sequence of row sums is A006128(n).

The number of times k appears in row n is A008284(n,k). - Franklin T. Adams-Watters, Jan 12 2006

The next level (row) gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent. See A128628. - Alford Arnold, Mar 27 2007

The 1's in the (flattened) sequence mark the start of a new row, the value that precedes the 1 equals the row number minus one. (I.e., the 1 preceded by a 0 is the start of row 1, the 1 preceded by a 6 is the start of row 7, etc.) - M. F. Hasler, Jun 06 2018

Also the maximum part in the n-th partition in graded lexicographic order (sum/lex, A193073). - Gus Wiseman, May 24 2020