A267849 Triangular array: T(n,k) is the 2-row creation rook number to place k rooks on a 3 x n board.
1, 1, 3, 1, 6, 12, 1, 9, 36, 60, 1, 12, 72, 240, 360, 1, 15, 120, 600, 1800, 2520, 1, 18, 180, 1200, 5400, 15120, 20160, 1, 21, 252, 2100, 12600, 52920, 141120, 181440, 1, 24, 336, 3360, 25200, 141120, 564480, 1451520, 1814400, 1, 27, 432, 5040, 45360, 317520, 1693440, 6531840, 16329600, 19958400
Offset: 0
Examples
The triangle T(n,k) begins in row n=0 with columns 0<=k<=n:
1
1 3
1 6 12
1 9 36 60
1 12 72 240 360
1 15 120 600 1800 2520
1 18 180 1200 5400 15120 20160
1 21 252 2100 12600 52920 141120 181440
1 24 336 3360 25200 141120 564480 1451520 1814400
1 27 432 5040 45360 317520 1693440 6531840 16329600 19958400
Links
- Jay Goldman and James Haglund, Generalized rook polynomials, J. Combin. Theory A 91 (2000), 509-530.
Crossrefs
Cf. A013610 (1-rook coefficients on the 3xn board), A121757 (2-rook coeffs. on the 2xn board), A013609 (1-rook coeffs. on the 2xn board), A013611 (1-rook coeffs. on the 4xn board), A008279 (2-rook coeffs. on the 1xn board), A082030 (row sums?), A049598 (column k=2), A007531 (column k=3 w/o factor 10), A001710 (diagonal?).
Formula
T(n,k) = T(n-1,k) + (k+2) T(n-1,k-1) subject to T(0,0)=1, T(n,k)=0 for n
Extensions
Triangle simplified (reversing rows, offset 0). - R. J. Mathar, May 03 2017
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
Comments
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.
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