# [seqfan] Non-Attacking Triangular Board Rooks, guess the recurrence

Ron Hardin rhhardin at att.net
Sat Aug 6 20:13:07 CEST 2011

```A series of antisymmetric recurrences for the columns, can it be guessed?

T(n,k)=Number of ways to arrange k nonattacking triangular rooks on an nXnXn
triangular grid

Empirical: minimum-n nonzero T(n,k) is at n=k+floor(k/2) and this
T(k+floor(k/2),k) is A002047((k-1)/2) for k odd.

Table starts
...1....0......0.......0........0........0.........0.........0........0.......0
...3....0......0.......0........0........0.........0.........0........0.......0
...6....3......0.......0........0........0.........0.........0........0.......0
..10...15......2.......0........0........0.........0.........0........0.......0
..15...45.....23.......0........0........0.........0.........0........0.......0
..21..105....127......18........0........0.........0.........0........0.......0
..28..210....468.....233........6........0.........0.........0........0.......0
..36..378...1352....1449......270........0.........0.........0........0.......0
..45..630...3310....6213.....3195......166.........0.........0........0.......0
..55..990...7190...20993....21273.....4902........28.........0........0.......0
..66.1485..14260...59943...101484....54771......4842.........0........0.......0
..78.2145..26330..150903...386052...382439....104448......2532........0.......0
..91.3003..45885..344323..1243899..1976455...1127473....140598......244.......0
.105.4095..76237..726033..3527469..8250687...8147469...2568288...120052.......0
.120.5460.121688.1434678..9035376.29309540..44813100..27060693..4373740...49620
.136.7140.187712.2685046.21297492.91705972.201616740.200826477.71690568.5227020

Empirical for columns:
T(n,1)=3*T(n-1,1)-3*T(n-2,1)+T(n-3,1)
T(n,2)=5*T(n-1,2)-10*T(n-2,2)+10*T(n-3,2)-5*T(n-4,2)+T(n-5,2)
T(n,3)=6*T(n-1,3)-14*T(n-2,3)+14*T(n-3,3)-14*T(n-5,3)+14*T(n-6,3)-6*T(n-7,3)+T(n-8,3)

T(n,4)=6*T(n-1,4)-12*T(n-2,4)+2*T(n-3,4)+27*T(n-4,4)-36*T(n-5,4)+36*T(n-7,4)-27*T(n-8,4)-2*T(n-9,4)+12*T(n-10,4)-6*T(n-11,4)+T(n-12,4)

T(n,5)=5*T(n-1,5)-5*T(n-2,5)-14*T(n-3,5)+30*T(n-4,5)+6*T(n-5,5)-50*T(n-6,5)+10*T(n-7,5)+44*T(n-8,5)-44*T(n-10,5)-10*T(n-11,5)+50*T(n-12,5)-6*T(n-13,5)-30*T(n-14,5)+14*T(n-15,5)+5*T(n-16,5)-5*T(n-17,5)+T(n-18,5)

Column 1 is A000217
Column 2 is A050534

Some solutions for n=5 k=3
......1..........0..........0..........0..........0..........0..........0..
.....0.0........0.0........0.1........0.0........0.1........0.0........0.0..
....0.0.0......1.0.0......1.0.0......0.1.0......0.0.0......0.1.0......0.0.1..
...0.0.1.0....0.0.1.0....0.0.0.0....1.0.0.0....0.0.1.0....0.0.0.1....0.1.0.0..
..0.1.0.0.0..0.1.0.0.0..0.0.0.1.0..0.0.0.0.1..1.0.0.0.0..1.0.0.0.0..0.0.0.1.0..

rhhardin at mindspring.com
rhhardin at att.net (either)

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