cp's OEIS Frontend

Let Theta(n) denote the set of norm values corresponding to all the partitions of n. The following results hold regarding this set: (i) Theta(n) is a subset of Theta(n+1); (ii) A prime p will appear as a norm only for partitions of n>=p; (iii) There exists a prime p not in Theta(n) for all n>=6; (iv) Let h(k) be the prime floor function which gives the greatest prime less than or equal to the k, then the prime p=h(n+1) does not belong to Theta(n); and (v) The primes not in the set Theta(n) are A000720(A000792(n)) - A000720(n). - Abhimanyu Kumar, Nov 25 2020

Column 0 of A119825.

From Wolfdieter Lang, Dec 08 2020: (Start)

The sequence b(n) = a(n-1), for n >= 1, and b(0) = 1, with o.g.f. Gb(x) = (1 - x - x^2 - x^3)*G(x), where G(x) = 1/(1 - 2*x - 2*x^2 - 2*x^3) generates A077835, is the INVERT transform of the tribonacci sequence {Trib(k+2)}_{k >= 1}, with Trib(n) = A000073(n). See the Bernstein and Sloane link for INVERT.

The proof that (1 - 2*x - 2*x^2 - 2*x^3) = (1 - x - x^2 - x^3)*(1 - Sum_{k = 1..M} Trib(k+2)*x^k), for M >= 3, up to terms starting with Trib(M+3)*x^{M+1} can be done by induction, using the tribonacci recurrence. Letting M -> infinity one obtains the o.g.f. of {b(n)}_{n>=0} from the one given by the INVERT transform.

The explicit form of b(n), for n >= 1, is given in terms of the partition array A048996 (M_0-multinomials) with the multivariate row polynomials with indeterminates {Trib(k+2)}_{k = 1..n}. See the example section instead of giving the general baroque partition formula. (End)

A060642 describes the ordered case.

Number of twice-partitions of n of length k. A twice-partition of n is a choice of a partition of each part in a partition of n. - Gus Wiseman, Mar 23 2018

Triangular array distributing the Bell numbers (A000110). The value associated with each partition is the product of A074664(k) for each part of size k, times the number of compositions associated with the partition (A048996 & A072881). The value for T(n,k) is the total of these values for each partition of n into k parts.

Calculating the appropriate weights can be done by "working backward". Suppose for example we know the weights for 1 through 6 and desire the weight for the partitions of seven: Substitute the weights for each partition value and multiply. For example, 7 = 4+3 so f([4,3]) = 6*2 = 12; adjusting for the number of permutations of [4,3] we now have 2*12 = 24. Continuing in this manner for each partition of seven and summing to 451 we now know all of the values except that associated with the partition [7] which must be 877 - 451 = 426.

From Mike Zabrocki: (Start)

Every set partition can be uniquely split into "atomic" set partitions or is itself already atomic.

{{1},{2},{3}} = {{1}}|{{1}}|{{1}}

{{1},{23}} = {{1}}|{{12}}

{{12},{3}} = {{12}}|{{1}}

{{13},{2}} is already atomic

{{123}} is already atomic

where this operation | is defined as {A1,...,Ar}|{B1,...,Bs} = {A1,...,Ar,B1+n,...,Bs+n}

where Bi+n = {bi1+n,bi2+n,...,bik+n} if Bi = {bi1,bi2,...,bik} and n = |A1|+|A2|+...+|Ar|. (End)

Subtriangle (n >= 1 and 1 <= k <= n) of triangle given by [0,1,1,2,1,3,1,4,1,5,1,6,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 03 2007

From Peter Bala, Aug 05 2014: (Start)

Let B(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + ... denote the o.g.f. for the Bell numbers A000110. Let f(x) = (B(x) - 1)/(x*B(x)) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + ..., the o.g.f. for the first column of this array. Then this array appears to be the Riordan array (f(x), x*f(x)).

If true, this gives the o.g.f. of the array as (B(x) - 1)/( x*(t + (1 - t)*B(x)) ) = 1 + (1 + t)*x + (2 + 2*t + t^2)*x^2 + ... and also the hockey-stick recurrence: T(n+1,k+1) = T(n,k) + T(n-1,k) + 2*T(n-2,k) + 6*T(n-3,k) + 22*T(n-4,k) + ..., n,k >= 1. (End)

This partition array is the member k=1 in the family M_0(k), with M_0(2)=M_0= A048996, M_0(3)= A134283, etc.

When read as partition array (tabf with sequence of row lengths given by the partition numbers A000041) in Abramowitz-Stegun order (see A117506 for the reference) a(n,k) is the characteristic partition array for the partition (1^n) of n.

The sums of row n are given in A000041(n), for n >= 1 (number of partitions).

A differently signed triangle is A047265.

One could add a column m = 0 starting at n = 0 with T(0, 0) = 1 and T(n, 0) = 0 otherwise, by including the empty partition with no parts.

For the weights w of positive integer numbers n see a comment in A339885. It is w(n) = -A010815(n), for n >= 0. Also w(n) = A257628(n), for n >= 1.

The weight of a composition is the one of the respective partition, obtained by the product of the weights of the parts.

That the row sums give the number of partitions follows from the pentagonal number theorem. See also the Apr 04 2013 conjecture in A000041 by Gary W. Adamson, and the hint for the proof by Joerg Arndt. The INVERT map of A = {1, 1, 0, 0, -5, -7, ...}, with offset 1, gives the A000041(n) numbers, for n >= 0.

If the above mentioned column for m = 0, starting at n = 0 is added this is an ordinary convolution triangle of the Riordan type R(1, f(x)), with f(x) = -(Product_{j>=1} (1 - x^j) - 1), generating {A257628(n)}{n>=0}. See the formulae below. - _Wolfdieter Lang, Feb 16 2021

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.

Partition number array M_0(3); the k=3 member in the family of a generalization of the multinomial number arrays M_0 = M_0(2) = A048996.

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].

The s2(3,n,m):=A035324(n,m) numbers (generalized Pascal triangle) are obtained by summing in row n all numbers with the same part number m. In the same manner the s2(2,n,m) = binomial(n-1,m-1) = A007318(n-1,m-1) numbers are obtained from the partition array M_0 = A048996.

(1-sqrt(1-4*x))/(2*x) = Sum_{k>=0} C(k)*x^k with C(n)=A000108(n) written as formal Product_{n>=1} (1 + a(n)*x^n).

This sequence connects the multinomial coefficients A036040 (M_3) with A127671 (M_5).

The numbers of terms in n-th row is the number of partitions A000041(n). The number of terms T(n, k) with equal values in the n-th row follow the rhythm of A008284(n).

Some row sums are [1, 0, 2, -5, 21, -104, 636, -4511, 36455, -330954, 3334390, -36914039].

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.

Partition number array M_0(3)= A134283 with each entry divided by the corresponding one of the partition number array M_0 = M_0(2) = A048996; in short M_0(3)/M_0.