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

%I A180401 #19 Jan 08 2025 10:39:13
%S A180401 1,0,1,0,1,1,0,4,4,1,0,36,33,10,1,0,576,480,148,20,1,0,14400,10960,
%T A180401 3281,483,35,1,0,518400,362880,103824,15552,1288,56,1,0,25401600,
%U A180401 16465680,4479336,663633,57916,2982,84,1,0,1625702400,981872640,253732096,36690816,3252624,181312,6216,120,1
%N A180401 Stirling-like sequence obtained from bipartite 0-1 tableaux.
%C A180401 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.
%H A180401 K. J. M. Gonzales, <a href="https://arxiv.org/abs/1008.4192">Enumeration of Restricted Permutation Pairs and Partitions Pairs via 0-1 Tableaux</a>, arXiv:1008.4192 [math.CO], 2010-2014.
%H A180401 A. de Medicis and P. Leroux, <a href="https://doi.org/10.4153/CJM-1995-027-x">Generalized Stirling Numbers, Convolution Formulae and p,q-Analogues</a>, Can. J. Math. 47 (1995), 474-499.
%F A180401 G.f.: sum_{all r=>0} C(n,k) x^r = prod_{all v+w=n,0<=v,w<=n-1} (x+vw)
%F A180401 Symm. f: C(n,k)=sum_{all 0 <=i_1<i_2<...<i_{n-k}<=n-1}
%F A180401 (i_1*(n-1)-i_1)*(i_2*(n-1)-i_2)*...*(i_{n-k}*(n-1)-i_{n-k})
%F A180401 Recurrences: Let C(n,k;r)=sum_{all 0 <=i_1<i_2<...<i_{n-k}<=n-1}
%F A180401 (i_1*(r+(n-1)-i_1))*(i_2*(r+(n-1)-i_2))*...*(i_{n-k}*(r+(n-1)-i_{n-k})). Then,
%F A180401 C(n,k)=C(n-1,k-1,1)+(n)C(n-1,k,1)
%e A180401 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
%o A180401 (R) ## Runs on R 2.7.1
%o A180401 ## Here, beta=r in recurrences
%o A180401 cnk<-function(n,k,beta=0){
%o A180401 alpha=0
%o A180401 as<-function(j){j}
%o A180401 bs<-function(j){j}
%o A180401 form.seq<-function(n,fcn){ss<-NULL;for(i in 0:n){ss<-c(ss,fcn(i))};ss}
%o A180401 seq.a<-form.seq(n+alpha+1,as)
%o A180401 seq.b<-form.seq(n+beta+1,bs)
%o A180401 v<-function(i){i}
%o A180401 w<-function(i){i}
%o A180401 if(n>k){
%o A180401 Atab<-combn(1:n-1,n-k)
%o A180401 Btab<-n-1-Atab+beta
%o A180401 Atab<-Atab+alpha
%o A180401 px<-NULL
%o A180401 for(i in 1:ncol(Atab)){
%o A180401 partial<-NULL
%o A180401 for(j in 1:nrow(Atab)){
%o A180401 partial<-c(partial,(v(seq.a[Atab[j,i]+1])*w(seq.b[Btab[j,i]+1])))
%o A180401 } # for(j in 1:nrow(Atab))
%o A180401 px<-c(px,prod(partial))
%o A180401 }# for(i in 1:ncol(Atab))
%o A180401 } # if(n>k)
%o A180401 if(n>k) x<-sum(px)
%o A180401 if(n==k) x=1
%o A180401 if(n<k) x=0
%o A180401 x
%o A180401 }
%o A180401 # Example
%o A180401 cnk(7,4)
%Y A180401 Cf. A000292, A080251.
%K A180401 nonn,tabl
%O A180401 1,8
%A A180401 _Ken Joffaniel M Gonzales_, Sep 02 2010, Sep 27 2010