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Square array read by descending antidiagonals of number of ways of dividing n*k labeled items into k unlabeled orders with n items in each order.
1

%I #13 Feb 19 2024 04:35:49

%S 1,1,2,1,12,6,1,120,360,24,1,1680,60480,20160,120,1,30240,19958400,

%T 79833600,1814400,720,1,665280,10897286400,871782912000,217945728000,

%U 239500800,5040,1,17297280,8892185702400,20274183401472000

%N Square array read by descending antidiagonals of number of ways of dividing n*k labeled items into k unlabeled orders with n items in each order.

%C T(p,k) = (pk)!/k! is divisible by p^k but not p^(k+1) for p prime; e.g., T(3,4) = 3^4*11*10*8*7*5*4*2*1 = 19958400.

%H Paolo Xausa, <a href="/A066991/b066991.txt">Table of n, a(n) for n = 1..820</a> (antidiagonals 1..40 of the array, flattened).

%F T(n,k) = (n*k)!/k!.

%e The array begins:

%e n\k| 1 2 3 4 ...

%e --------------------------------------------------------

%e 1 | 1, 1, 1, 1, ...

%e 2 | 2, 12, 120, 1680, ...

%e 3 | 6, 360, 60480, 19958400, ...

%e 4 | 24, 20160, 79833600, 871782912000, ...

%e 5 | 120, 1814400, 217945728000, 101370917007360000, ...

%e ...

%t Table[((n-k+1)*k)!/k!, {n, 10}, {k, n, 1, -1}] (* _Paolo Xausa_, Feb 19 2024 *)

%Y Rows include A000012, A001813, A064350.

%Y Columns include A000142, A002674, A065961.

%Y Cf. A060538, A060539, A060540.

%K nonn,tabl

%O 1,3

%A _Henry Bottomley_, Feb 01 2002

%E Edited by _Paolo Xausa_, Feb 19 2024