cp's OEIS Frontend

a(n) = degree of Kac determinant at level n as polynomial in the conformal weight (called h). (Cf. C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Vol. 2, p. 533, eq.(98); reference p. 643, Cambridge University Press, (1989).) - Wolfdieter Lang

Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that from any part z > 1 one can take an element of amount 1 in one way only. That means z is composed of z unlabeled parts of amount 1, i.e. z = 1 + 1 + ... + 1. E.g., for n=3 to n=2 we have a(3) = 6 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. For the case of z composed by labeled elements, z = 1_1 + 1_2 + ... + 1_z, see A066186. - Thomas Wieder, May 20 2004

Number of times a derivative of any order (not 0 of course) appears when expanding the n-th derivative of 1/f(x). For instance (1/f(x))'' = (2 f'(x)^2-f(x) f''(x)) / f(x)^3 which makes a(2) = 3 (by counting k times the k-th power of a derivative). - Thomas Baruchel, Nov 07 2005

Starting with offset 1, = the partition triangle A008284 * [1, 2, 3, ...]. - Gary W. Adamson, Feb 13 2008

Starting with offset 1 equals A000041: (1, 1, 2, 3, 5, 7, 11, ...) convolved with A000005: (1, 2, 2, 3, 2, 4, ...). - Gary W. Adamson, Jun 16 2009

Apart from initial 0 row sums of triangle A066633, also the Möbius transform is A085410. - Gary W. Adamson, Mar 21 2011

More generally, the total number of parts >= k in all partitions of n equals the sum of k-th largest parts of all partitions of n. In this case k = 1. Apart from initial 0 the first column of A181187. - Omar E. Pol, Feb 14 2012

Row sums of triangle A221530. - Omar E. Pol, Jan 21 2013

From Omar E. Pol, Feb 04 2021: (Start)

a(n) is also the total number of divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.

Apart from initial zero this is also as follows:

Convolution of A000005 and A000041.

Convolution of A006218 and A002865.

Convolution of A341062 and A000070.

Row sums of triangles A221531, A245095, A339258, A340525, A340529. (End)

Number of ways to choose a part index of an integer partition of n, i.e., partitions of n with a selected position. Selecting a part value instead of index gives A000070. - Gus Wiseman, Apr 19 2021

Here the j-th "region" of the set of partitions of n (or more simply the j-th "region" of n) is defined to be the first h elements of the sequence formed by the smallest parts in nonincreasing order of the partitions of the largest part of the j-th partition of n, with the list of partitions in colexicographic order, where h = j - i, and i is the index of the previous partition of n whose largest part is greater than the largest part of the j-th partition of n, or i = 0 if such previous largest part does not exist. The largest part of the j-th region of n is A141285(j) and the number of parts is h = A194446(j).

Some properties of the regions of n:

- The number of regions of n equals the number of partitions of n (see A000041).

- The set of regions of n contain the sets of regions of all positive integers previous to n.

- The first j regions of n are also first j regions of all integers greater than n.

- The sums of all largest parts of all regions of n equals the total number of parts of all regions of n. See A006128(n).

- If T(j,1) is a record in the sequence then the leading diagonals of triangle formed by the first j rows give the partitions of n (see example).

- The rank of a region is the largest part minus the number of parts (see A194447).

- The sum of all ranks of the regions of n is equal to zero.

How to make a diagram of the regions and partitions of n: in the first quadrant of the square grid we draw a horizontal line {[0, 0],[n, 0]} of length n. Then we draw a vertical line {[0, 0],[0, p(n)]} of length p(n) where p(n) is the number of partitions of n. Then, for j = 1..p(n), we draw a horizontal line {[0, j],[g, j]} where g = A141285(j) is the largest part of the j-th partition of n, with the list of partitions in colexicographic order. Then, for n = 1 .. p(n), we draw a vertical line from the point [g,j] down to intercept the next segment in a lower row. So we have a number of closed regions. Then we divide each region of n in horizontal rectangles with shorter sides = 1. We can see that in the original rectangle of area n*p(n) each row contains a set of rectangles whose areas are equal to the parts of one of the partitions of n. Then each region of n is labeled according to the position of its largest part on axis "y". Note that each region of n is similar to a mirror version of the Young diagram of one of the partitions of s, where s is the sum of all parts of the region. See the illustrations of the seven regions of 5 in the Links section.

Note that if row j of triangle contains parts of size 1 then the parts of row j are the smallest parts of all partitions of T(j,1), (see A046746), and also T(j,1) is a record in the sequence and also j is the number of partitions of T(j,1), (see A000041). Otherwise, if row j does not contain parts of size 1 then the parts of row j are the emergent parts of the next record in the sequence (see A183152). Row j is also the partition of A186412(j).

Also triangle read by rows in which row r lists the parts of the last section of the set of partitions of r, ordered by regions, such that the previous parts to the part of size r are the emergent parts of the partitions of r (see A138152) and the rest are the smallest parts of the partitions of r (see example). - Omar E. Pol, Apr 28 2012

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)

Triangle T(n,k) read by rows in which, from rows 1..n, if r = T(n,k) is a record in the sequence then the set of positive integers in every row (from 1 to n) is called a “region” of r. Note that n, the number of regions of r is also the number of partitions of r. The consecutive records "r" are the natural numbers A000027. The triangle has the property that, for rows n..1, the diagonals (without the zeros) are also the partitions of r, in juxtaposed reverse-lexicographical order. Note that a record "r" is the initial term of a row if such row contains 1’s. If T(n,k) is a record in the sequence then A000041(T(n,k)) = n. Note that if T(n,k) < 2 is not the last term of the row n then T(n,k+1) = T(n,k). The union of the rows that contain 1's gives A182715.

Also number of partitions of n in which the largest part occurs an odd number of times. Example: a(5)=6 because we have [5],[4,1],[3,2],[3,1,1],[2,1,1,1] and [1,1,1,1,1] ([2,2,1] does not qualify). - Emeric Deutsch, Apr 04 2006

Also triangle read by rows: T(j,k) = sum of all parts in the k-th region of the last section of the set of partitions of j. See Example section. For more information see A135010. - Omar E. Pol, Nov 26 2011

For the definition of "region" see A206437. - Omar E. Pol, Aug 19 2013

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

Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n. For more information see A182699.

Also total sum of parts of the regions that do not contain 1 as a part in the last section of the set of partitions of n (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

Also number of compositions of n+1 with unique largest part. - Vladeta Jovovic, Apr 03 2005