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A022917
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Multinomial coefficient n!/ ([n/4]!, [(n+1)/4]!, [(n+2)/4]!, [(n+3)/4]!).
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4
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1, 1, 2, 6, 24, 60, 180, 630, 2520, 7560, 25200, 92400, 369600, 1201200, 4204200, 15765750, 63063000, 214414200, 771891120, 2933186256, 11732745024, 41064607584, 150570227808, 577185873264, 2308743493056, 8245512475200
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OFFSET
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0,3
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COMMENTS
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Multinomial coefficients(TOP, BOTTOM), where TOP = n, BOTTOM = ( a b c d ) where a = [ n/4 ], b = [ n/4 ], c = [ n/4 ], d = [ n/4 ] if n==0 mod 4; a = [ n/4 ], b = [ n/4 ], c = [ n/4 ], d = [ n/4 ]+1 if n==1 mod 4; a = [ n/4 ], b = [ n/4 ], c = [ n/4 ]+1, d = [ n/4 ]+1 if n==2 mod 4; a = [ n/4 ], b = [ n/4 ]+1, c = [ n/4 ]+1, d = [ n/4 ]+1 if n==3 mod 4.
Number of permutation patterns modulo 4. This matches the multinomial formula. - Olivier Gérard, Feb 25 2011
Also the number of permutations of n elements where p(k-4) < p(k) for all k. - Joerg Arndt, Jul 23 2011
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LINKS
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FORMULA
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Conjecture: -(126*n-1)*(n+3)*(n+2)*(n+1)*a(n) +4*(-208*n^4-51*n^3+793*n^2+376*n-6)*a(n-1) +16*(334*n^3-114*n^2-5*n-369)*a(n-2) +64*(n-2)*(334*n^2+264*n-213)*a(n-3) +768*(n-2)*(n-3)*(42*n^2+153*n+113)*a(n-4) +1024*(208*n+177)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 06 2015
Recurrence: (n+1)*(n+2)*(n+3)*(4*n^3 + 8*n^2 + 4*n - 1)*a(n) = 8*(2*n^2 + 2*n - 1)*(n^3 + 4*n^2 + 7*n + 3)*a(n-1) + 16*(n-1)*(4*n^4 + 16*n^3 + 16*n^2 + 18*n + 15)*a(n-2) + 128*(n-2)*(n-1)*(2*n^3 + 14*n^2 + 31*n + 18)*a(n-3) + 256*(n-3)*(n-2)*(n-1)*(4*n^3 + 20*n^2 + 32*n + 15)*a(n-4).
a(n) ~ 2^(2*n + 5/2) / (Pi*n)^(3/2). (End)
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EXAMPLE
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Starting from n=5, several permutations have the same pattern mod 4. Both (4,1,5,2,3) and (4,5,1,2,3) have pattern (0,1,1,2,3) modulo 4.
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MAPLE
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n!/floor(n/4)!/floor((n+1)/4)!/floor((n+2)/4)!/floor((n+3)/4)! ;
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MATHEMATICA
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Table[ n!/(Quotient[n, 4]!*Quotient[n + 1, 4]!*Quotient[n + 2, 4]!*
Quotient[n + 3, 4]!), {n, 0, 30}]
Table[n!/Times@@(Floor[Range[n, n+3]/4]!), {n, 0, 30}] (* Harvey P. Dale, May 30 2018 *)
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PROG
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(PARI) {a(n)= if(n<0, 0, n!/(n\4)!/((n+1)\4)!/((n+2)\4)!/((n+3)\4)!)} /* Michael Somos, Jun 20 2007 */
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CROSSREFS
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Cf. A001405 (permutation patterns mod 2).
Cf. A022916 (permutation patterns mod 3).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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