A219237 -id:A219237 - cp's OEIS frontend

A232539 Triangle read by rows: T(n,k) = number of partitions of n into at most four parts in which the largest part is equal to k, 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 2, 3, 2, 1, 1, 0, 0, 1, 3, 3, 2, 1, 1, 0, 0, 1, 3, 4, 3, 2, 1, 1, 0, 0, 0, 3, 4, 4, 3, 2, 1, 1, 0, 0, 0, 2, 5, 5, 4, 3, 2, 1, 1, 0, 0, 0, 1, 4, 6, 5, 4, 3, 2, 1, 1, 0, 0, 0, 1, 4, 6, 7, 5, 4, 3, 2, 1, 1

Offset: 0

L. Edson Jeffery, Jan 02 2014

Also number of partitions of n into k parts with parts in the range 1..4.

			Triangle T{n,k} begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1, 1;
  0, 0, 2, 2, 1, 1;
  0, 0, 2, 3, 2, 1, 1;
  0, 0, 1, 3, 3, 2, 1, 1;
  0, 0, 1, 3, 4, 3, 2, 1, 1;
  0, 0, 0, 3, 4, 4, 3, 2, 1, 1;
  ...

Louis Comtet, Advanced Combinatorics, Reidel (1974).

Cf. A001400 (row sums), A219237, A233292 (row partial sums), A145362 (parts <=2), A339884 (parts <=3).

maxp := 4 :
gf := 1/mul(1-u*t^i,i=1..maxp) :
for n from 0 to 13 do
    for m from 0 to n do
        coeftayl(gf,t=0,n) ;
        coeftayl(%,u=0,m) ;
        printf("%d ",%);
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 27 2025

G.f.: 1/((1-u*t)*(1-u*t^2)*(1-u*t^3)*(1-u*t^4)). - [Comtet p. 97 [2c]]. - R. J. Mathar, May 27 2025

A219238 Coefficient table for the first differences of table A047971: Coefficients of the difference of Gauss polynomials [n+3,3]_q - [n+2,3]_q.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 2, 2, 2, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 0

Wolfdieter Lang, Dec 06 2012

The row lengths sequence is A016777 (3*n+1). The sum for row n is A000217(n+1) = binomial(n+2,2).
The coefficients of the Gauss polynomial [n+3,3]_q are given in A047971.
a(n,k) = [q^k]([n+3,3]_q - [n+2,3]_q). One can use the identity [n+3,3]_q - [n+2,3]_q = q^n*[n+2,2]_q (see the Andrews reference given in A047971, p. 35, (3.3.3)). Therefore, the present array is obtained from A008967 after a shift of row n by n units to the right, inserting zeros for the first n entries.
The o.g.f. of the row polynomials in q of degree 3*n is 1/((1-q)*(1-q^2)*(1-q^3)) (multiply the o.g.f. of A047971 by (1-z)). a(n,k) determines therefore the number of partitions of k with precisely n parts, each <= 3. Alternatively, a(n,k) determines the number of partitions of k with at most 3 parts, with each part <= n but not each part <= (n-1), i.e., part n, maybe more than once, is present besides possibly smaller ones.

			The table a(n,k) begins:
n\k 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18...
0:  1
1:  0  1  1  1
2:  0  0  1  1  2  1  1
3:  0  0  0  1  1  2  2  2  1  1
4:  0  0  0  0  1  1  2  2  3  2  2  1  1
5:  0  0  0  0  0  1  1  2  2  3  3  3  2  2  1  1
6:  0  0  0  0  0  0  1  1  2  2  3  3  4  3  3  2  2  1  1
...
Row n=1 is 0,1,1,1 because [3,2]_q = 1 + q + q^2 and the coefficient of q^{-1} is 0, the one of q^0 is 1, the one of q^1 is 1 and the one of q^2 is 1. A shift of row n=1 of A008967 by one unit to the right.
a(n,k) = 0 if n > k because a partition of k never has more than k parts.
a(n,k) = 0 if k > 3*n because there is no partition of 3*n+m, with m >= 1, and exactly n parts, each <= 3.
a(2,4) = 2 because the partitions of 4 with 2 parts are 1,3 and 2,2, and the parts in both are <= 3.
a(2,4) = 2 because the partitions of 4 with number of parts <= 3, each <= 2, are 2,2 and 1,1,2, and part 2 is present in both of them. Note the conjugacy of partitions 1,3 and 1,1,2.

Cf. A047971, A008967 (with shifted rows).

a(n,k) = [q^k]([n+3,3]_q - [n+2,3]_q), = [q^(k-n)] [n+2,2]_q , n >= 0, 0 <= k <= 3*n. For the Gauss polynomial (q-binomial) [n+m,m]_q = [m+n,n]_q see a comment on A219237 where also the Andrews reference and a link to Mathworld is found.

A089789 Number of irreducible factors of Gauss polynomials.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 1, 2, 2, 1, 0, 0, 3, 3, 4, 3, 3, 0, 0, 1, 3, 3, 3, 3, 1, 0, 0, 3, 3, 5, 4, 5, 3, 3, 0, 0, 2, 4, 4, 5, 5, 4, 4, 2, 0, 0, 3, 4, 6, 5, 7, 5, 6, 4, 3, 0, 0, 1, 3, 4, 5, 5, 5, 5, 4, 3, 1, 0, 0, 5, 5, 7, 7, 9, 7, 9, 7, 7, 5, 5, 0, 0, 1, 5, 5, 6, 7, 7, 7, 7, 6, 5, 5, 1, 0

Offset: 0

Paul Boddington, Jan 09 2004

T(n,k) is the number of irreducible factors of the (separable) polynomial [n]!/([k]![n-k]!). Here [n]! denotes the product of the first n quantum integers, the n-th quantum integer being defined as (1-q^n)/(1-q).

T(n,k) gives the number of positive integers m <= n such that (n mod m) < (k mod m). - Tom Edgar, Aug 21 2014

			The triangle T(n,k) begins:
n\k  0  1  2  3  4  5  6  7  8  9  10  11  12  13 ...
0:   0
1:   0  0
2:   0  1  0
3:   0  1  1  0
4:   0  2  2  2  0
5:   0  1  2  2  1  0
6:   0  3  3  4  3  3  0
7:   0  1  3  3  3  3  1  0
8:   0  3  3  5  4  5  3  3  0
9:   0  2  4  4  5  5  4  4  2  0
10:  0  3  4  6  5  7  5  6  4  3   0
11:  0  1  3  4  5  5  5  5  4  3   1   0
12:  0  5  5  7  7  9  7  9  7  7   5   5   0
13:  0  1  5  5  6  7  7  7  7  6   5   5   1   0
... Formatted by _Wolfdieter Lang_, Dec 07 2012
T(8,3) equals the number of irreducible factors of (1-q^8)(1-q^7)(1-q^6)/((1-q^3)(1-q^2)(1-q)), which is a product of 5 cyclotomic polynomials in q, namely the 2nd, 4th, 6th, 7th and 8th. Thus T(8,3)=5.

Stan Wagon and Herbert S. Wilf, When are subset sums equidistributed modulo m?, The Electronic Journal of Combinatorics, Vol. 1, 1994 (#R3).

Cf. A008967, A047971, A219237, A000005.

T(n, k) = T(n-1, k-1) + d(n) - d(k), where d(n) is the number of divisors of n.

A233292 Triangle read by rows: T(n,k) is the number of partitions of n into at most four parts in which no part exceeds k, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 4, 5, 0, 0, 2, 4, 5, 6, 0, 0, 2, 5, 7, 8, 9, 0, 0, 1, 4, 7, 9, 10, 11, 0, 0, 1, 4, 8, 11, 13, 14, 15, 0, 0, 0, 3, 7, 11, 14, 16, 17, 18, 0, 0, 0, 2, 7, 12, 16, 19, 21, 22, 23, 0, 0, 0, 1, 5, 11, 16, 20, 23, 25, 26, 27, 0, 0, 0, 1, 5, 11, 18, 23, 27, 30, 32, 33, 34

Offset: 0

L. Edson Jeffery, Jan 02 2014

Transpose of table A219237.

			Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 2, 3;
0, 1, 3, 4, 5;
0, 0, 2, 4, 5,  6;
0, 0, 2, 5, 7,  8,  9;
0, 0, 1, 4, 7,  9, 10, 11;
0, 0, 1, 4, 8, 11, 13, 14, 15; ...

Cf. A001400 (main diagonal), A219237, A232539.

T(n,k) = Sum_{j=0..k} A232539(n,j).

cp's OEIS Frontend

A232539 Triangle read by rows: T(n,k) = number of partitions of n into at most four parts in which the largest part is equal to k, 0 <= k <= n.

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A219238 Coefficient table for the first differences of table A047971: Coefficients of the difference of Gauss polynomials [n+3,3]_q - [n+2,3]_q.

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A089789 Number of irreducible factors of Gauss polynomials.

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A233292 Triangle read by rows: T(n,k) is the number of partitions of n into at most four parts in which no part exceeds k, 0 <= k <= n.

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