A180401 Stirling-like sequence obtained from bipartite 0-1 tableaux.
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
Examples
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
Links
- 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.
Programs
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R
## 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
Formula
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)
Comments