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Number of permutations of length n containing exactly 2 occurrences of 123 and 2 occurrences of 132.
1

%I #26 Feb 29 2020 13:37:43

%S 0,0,0,0,1,6,26,94,306,934,2732,7752,21488,58432,156288,411904,

%T 1071104,2750976,6984704,17545216,43634688,107511808,262602752,

%U 636223488,1529741312,3652059136,8660975616,20412104704,47826599936,111446851584,258360737792,596044152832

%N Number of permutations of length n containing exactly 2 occurrences of 123 and 2 occurrences of 132.

%H Alois P. Heinz, <a href="/A224288/b224288.txt">Table of n, a(n) for n = 0..1000</a>

%H B. Nakamura, <a href="http://arxiv.org/abs/1301.5080">Approaches for enumerating permutations with a prescribed number of occurrences of patterns</a>, arXiv 1301.5080 [math.CO], 2013.

%H B. Nakamura, <a href="http://www.math.rutgers.edu/~bnaka/GWILF2/F123n132">A Maple package for enumerating n-permutations with r occurrences of the pattern 123 and s occurrences of the pattern 132</a> [Broken link]

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).

%F G.f.: -(2*x^5+6*x^4-6*x^3+6*x^2-4*x+1)*x^4/(2*x-1)^5. - _Alois P. Heinz_, Apr 03 2013

%F a(n) = 2^(-11+n)*(1504-994*n+219*n^2-18*n^3+n^4) for n>4. - _Colin Barker_, Apr 14 2013

%e a(4) = 1: (1,2,4,3).

%e a(5) = 6: (2,3,5,1,4), (2,3,5,4,1), (2,5,1,3,4), (3,1,4,5,2), (4,1,2,5,3), (5,1,2,4,3).

%p # Programs can be obtained from the Nakamura link

%t Join[{0, 0, 0, 0, 1}, LinearRecurrence[{10, -40, 80, -80, 32}, {6, 26, 94, 306, 934}, 27]] (* _Jean-François Alcover_, Feb 29 2020 *)

%Y Cf. A000079, A001787, A001815, A046718, A001793.

%K nonn,easy

%O 0,6

%A _Brian Nakamura_, Apr 03 2013