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 A100749 Triangle read by rows: T(n,k)=number of 231- and 312-avoiding permutations of [n] having k fixed points. 1
 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 2, 0, 5, 0, 1, 0, 8, 0, 7, 0, 1, 4, 0, 18, 0, 9, 0, 1, 0, 20, 0, 32, 0, 11, 0, 1, 8, 0, 56, 0, 50, 0, 13, 0, 1, 0, 48, 0, 120, 0, 72, 0, 15, 0, 1, 16, 0, 160, 0, 220, 0, 98, 0, 17, 0, 1, 0, 112, 0, 400, 0, 364, 0, 128, 0, 19, 0, 1, 32, 0, 432, 0, 840, 0, 560, 0, 162, 0, 21, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Also number of compositions of n having k odd parts. Example: T(3,1)=3 because we have 3=2+1=1+2. Row n has n+1 terms. Sum of row n is 2^(n-1) (A000079(n-1)) for n>0. LINKS Alois P. Heinz, Rows n = 0..140, flattened T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, Annals of Combinatorics, 6, 2002, 407-418 (Theorem 2.8). FORMULA T(n, k)=2^[(n-k-2)/2]*[(n+3k)/(n-k)]*binomial((n+k-2)/2, k) if k x+y, %, [`if`(irem(j, 2)=1, 0, [][]), T(n-j)], 0) od; %[] fi end: seq (T(n), n=0..20); # Alois P. Heinz, Nov 06 2012 MATHEMATICA t[n_, k_] := Which[k == n, 1, k

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Last modified February 2 19:34 EST 2023. Contains 360024 sequences. (Running on oeis4.)