A080251 -id:A080251 - cp's OEIS frontend

A080416 Stirling-like number triangle defined by paired decomposition of C(n+3,3) = A000292.

1, 1, 1, 1, 4, 1, 1, 12, 10, 1, 1, 32, 67, 20, 1, 1, 80, 376, 252, 35, 1, 1, 192, 1909, 2560, 742, 56, 1, 1, 448, 9094, 22928, 12346, 1848, 84, 1, 1, 1024, 41479, 189120, 177599, 46912, 4074, 120, 1, 1, 2304, 183412, 1472704, 2318149

Offset: 0

Paul Barry, Feb 17 2003

Note that the Stirling numbers of the second kind are generated in a similar fashion by decomposing the triangular numbers C(n+2,2) as {1}, {1,2}, {1,2,3}, .... The defining sequence A000292 appears as the subdiagonal when the triangle is arranged in lower-triangular form. The second column is A001787.

Gives the number of ways to construct pairs of k-block partitions from 1 to n such that the sum of the minima of the i-th block of the first partition and the (k-i+1)th block of the second partition is n+1. - Ken Joffaniel M Gonzales, Jun 13 2010

			Rows are
  {1},
  {1,  1},
  {1,  4,  1},
  {1, 12, 10,  1},
  {1, 32, 67, 20,  1},
  ...

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013-2014.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.

Cf. A000292, A001787, A008277, A080251.

s[b_, n_, k_] := s[b, n, k] = Which[n==k==0, 1, n==0, 0, k==0, 0, True, s[b+1, n-1, k-1] + k*b*s[b, n-1, k]]
Table[s[0, n+2, k+2], {n, 0, 10}, {k, 0, n}] // Flatten
(* a specialization of equation (9) in the Corcino et al. paper *)
(* Mikhail Lavrov, Oct 12 2022 *)
T[ n_, k_] := If[n < 0, 0, SeriesCoefficient[x^k / Product[1 + x*(Floor[j/2] + 1)*(Floor[j/2] - k - 1), {j,0,k}], {x,0,n}]]; (* Michael Somos, Oct 12 2022 *)

{T(n, k) = if(n<0, 0, polcoeff(x^k / prod(j=0, k, 1 + x*(j\2 + 1)*(j\2 - k - 1) + x*O(x^n)), n))}; /* Michael Somos, Oct 12 2022 */

Columns are generated as follows: Display C(n+3, 3) as row sums of the triangle A080251, or {1}, {2, 2}, {3, 3, 4}, {4, 4, 6, 6}, {5, 5, 8, 8, 9}, ... The columns are then generated by 1/(1-x), 1/(1-2x)^2, 1/((1-3x)^2*(1-4x)), 1/((1-4x)^2*(1-6x)^2), etc.

A180401 Stirling-like sequence obtained from bipartite 0-1 tableaux.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 4, 1, 0, 36, 33, 10, 1, 0, 576, 480, 148, 20, 1, 0, 14400, 10960, 3281, 483, 35, 1, 0, 518400, 362880, 103824, 15552, 1288, 56, 1, 0, 25401600, 16465680, 4479336, 663633, 57916, 2982, 84, 1, 0, 1625702400, 981872640, 253732096, 36690816, 3252624, 181312, 6216, 120, 1

Offset: 1

Ken Joffaniel M Gonzales, Sep 02 2010, Sep 27 2010

Gives the number of ways to construct pairs of permutations of an n-element set into k cycles such that the sum of the minima of the i-th cycle of the first permutation and the (k-i+1)-th cycle of the second permutation is n+1.

			For n=6, C(6,0)=0, C(6,1)=0, C(6,2)=1, C(6,3)=32, C(6,4)=67, C(6,5)=20, C(6,6)=1

K. J. M. Gonzales, Enumeration of Restricted Permutation Pairs and Partitions Pairs via 0-1 Tableaux, arXiv:1008.4192 [math.CO], 2010-2014.
A. de Medicis and P. Leroux, Generalized Stirling Numbers, Convolution Formulae and p,q-Analogues, Can. J. Math. 47 (1995), 474-499.

Cf. A000292, A080251.

## Runs on R 2.7.1
## Here, beta=r in recurrences
cnk<-function(n,k,beta=0){
alpha=0
as<-function(j){j}
bs<-function(j){j}
form.seq<-function(n,fcn){ss<-NULL;for(i in 0:n){ss<-c(ss,fcn(i))};ss}
seq.a<-form.seq(n+alpha+1,as)
seq.b<-form.seq(n+beta+1,bs)
v<-function(i){i}
w<-function(i){i}
if(n>k){
Atab<-combn(1:n-1,n-k)
Btab<-n-1-Atab+beta
Atab<-Atab+alpha
px<-NULL
for(i in 1:ncol(Atab)){
partial<-NULL
for(j in 1:nrow(Atab)){
partial<-c(partial,(v(seq.a[Atab[j,i]+1])*w(seq.b[Btab[j,i]+1])))
} # for(j in 1:nrow(Atab))
px<-c(px,prod(partial))
}# for(i in 1:ncol(Atab))
} # if(n>k)
if(n>k) x<-sum(px)
if(n==k) x=1
if(n

G.f.: sum_{all r=>0} C(n,k) x^r = prod_{all v+w=n,0<=v,w<=n-1} (x+vw)

Symm. f: C(n,k)=sum_{all 0 <=i_1

(i_1*(n-1)-i_1)*(i_2*(n-1)-i_2)*...*(i_{n-k}*(n-1)-i_{n-k})

Recurrences: Let C(n,k;r)=sum_{all 0 <=i_1

(i_1*(r+(n-1)-i_1))*(i_2*(r+(n-1)-i_2))*...*(i_{n-k}*(r+(n-1)-i_{n-k})). Then,

C(n,k)=C(n-1,k-1,1)+(n)C(n-1,k,1)

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cp's OEIS Frontend

A080416 Stirling-like number triangle defined by paired decomposition of C(n+3,3) = A000292.

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