A143171 -id:A143171 - cp's OEIS frontend

A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle).

1, 1, 1, 3, 1, 1, 15, 3, 1, 1, 1, 105, 15, 3, 3, 1, 1, 1, 945, 105, 15, 9, 15, 3, 1, 3, 1, 1, 1, 10395, 945, 105, 45, 105, 15, 9, 3, 15, 3, 1, 3, 1, 1, 1, 135135, 10395, 945, 315, 225, 945, 105, 45, 15, 9, 105, 15, 9, 3, 1, 15, 3, 1, 3, 1, 1, 1, 2027025, 135135, 10395, 2835

Offset: 1

Wolfdieter Lang, Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M32hat(-1;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
If M32hat(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-1):= A144270(n,m).

			a(4,3)= 1 = |S2(-1,2,1)|^2. The relevant partition of 4 is (2^2).
[1]; [1,1]; [3,1,1]; [15,3,1,1,1]; [105,15,3,3,1,1,1]; ... [From _Wolfdieter Lang_, Oct 23 2008]

Wolfdieter Lang, First 10 rows of the array and more.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A144271 (M32hat(-2) array).

a(n,k)= product(|S2(-1,j,1)|^e(n,k,j),j=1..n) with |S2(-1,n,1)|= A001147(n-1) = (2*n-3)(!^2) (2-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

Formally a(n,k)= 'M32(-1)/M3' = 'A143171/A036040' (elementwise division of arrays).

Corrected all entries. Wolfdieter Lang, Oct 23 2008

A157401 A partition product of Stirling_2 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 9, 12, 15, 1, 25, 60, 75, 105, 1, 75, 330, 450, 630, 945, 1, 231, 1680, 3675, 4410, 6615, 10395, 1, 763, 9408, 30975, 41160, 52920, 83160, 135135, 1, 2619, 56952, 233415, 489510, 555660, 748440, 1216215

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 1,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143171.
Same partition product with length statistic is A001497.
Diagonal a(A000217) = A001147.
Row sum is A001515.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157402, A157403, A157404, A157405

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(2*j - 1).

A143172 Partition number array, called M32(-2), related to A004747(n,m) = |S2(-2;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 10, 6, 1, 80, 40, 12, 12, 1, 880, 400, 200, 100, 60, 20, 1, 12320, 5280, 2400, 1000, 1200, 1200, 120, 200, 180, 30, 1, 209440, 86240, 36960, 28000, 18480, 16800, 7000, 4200, 2800, 4200, 840, 350, 420, 42, 1, 4188800, 1675520, 689920, 492800, 224000, 344960

Offset: 1

Wolfdieter Lang, Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(-2;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+1)-ary trees if the outdegree is r>=0.
If M32(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle A004747(n,m)= |S2(-2;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

			a(4,3)=12. The relevant partition of 4 is (2^2). The 12 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are binary because r=1 vertices are binary (2-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4 versions due to the two binary root vertices.

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A143171 (M32(-1) array), A143173 (M32(-3) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-2,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-2,j,1)|^e(n,k,j),j=1..n), with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

cp's OEIS Frontend

A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle).

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A157401 A partition product of Stirling_2 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

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A143172 Partition number array, called M32(-2), related to A004747(n,m) = |S2(-2;n,m)| (generalized Stirling triangle).

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