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A362563 Triangle T(n, k) read by rows, where T(n, k) is the number of {123,132}-avoiding parking functions of size n with k active sites, for 2 <= k <= n+1.

%I A362563 #21 Apr 26 2023 15:39:32
%S A362563 1,1,2,1,3,4,3,5,8,8,8,14,17,20,16,24,40,49,50,48,32,75,123,147,151,
%T A362563 136,112,64,243,393,465,473,432,352,256,128,808,1294,1519,1540,1409,
%U A362563 1176,880,576,256,2742,4358,5087,5144,4721,3986,3088,2144,1280,512
%N A362563 Triangle T(n, k) read by rows, where T(n, k) is the number of {123,132}-avoiding parking functions of size n with k active sites, for 2 <= k <= n+1.
%C A362563 Consider a parking function of size n that avoids both 123 and 132.
%C A362563 Such a parking function can be represented as a labeled Dyck path (using steps N = (0, 1) and E = (1, 0) staying weakly above y = x), where the north steps are labeled with 1, 2, ..., n, and where consecutive north steps have increasing labels.
%C A362563 An active site is a point where the parking function's corresponding Dyck path touches y = x.
%C A362563 T(n, k) is the number of parking functions of size n with exactly k active sites.
%H A362563 Ayomikun Adeniran and Lara Pudwell, <a href="https://doi.org/10.54550/ECA2023V3S3R17">Pattern avoidance in parking functions</a>, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
%F A362563 T(n, k) = 0 if k < 2 or k > n+1
%F A362563 T(1, 2) = T(2, 2) = 1.
%F A362563 T(2, 3) = 2.
%F A362563 For n > 2, T(n, k) = 2*T(n-1, k-1) + Sum_{j=k-1..n-1} T(n-2, j).
%F A362563 T(n, n+1) = A000079(n-1).
%F A362563 Sum_{k=2..n+1} T(n, k) = T(n+2, 2) = A000958(n+1).
%e A362563 Triangle T(n, k) begins:
%e A362563      1;
%e A362563      1,    2;
%e A362563      1,    3,    4;
%e A362563      3,    5,    8,    8;
%e A362563      8,   14,   17,   20,   16;
%e A362563     24,   40,   49,   50,   48,   32;
%e A362563     75,  123,  147,  151,  136,  112,   64;
%e A362563    243,  393,  465,  473,  432,  352,  256,  128;
%e A362563    808, 1294, 1519, 1540, 1409, 1176,  880,  576,  256;
%e A362563   2742, 4358, 5087, 5144, 4721, 3986, 3088, 2144, 1280, 512;
%e A362563   ...
%e A362563 The eight {123,132}-avoiding parking functions of size 3 are 211, 212, 213, 221, 231, 311, 312, and 321.
%e A362563 In block notation:
%e A362563    211 is {2,3},{1},{}  -> NNENEE, which has 2 active sites;
%e A362563    212 is {2},{1, 3},{} -> NENNEE, which has 3 active sites;
%e A362563    213 is {2},{1},{3}   -> NENENE, which has 4 active sites;
%e A362563    221 is {3},{1,2},{}  -> NENNEE, which has 3 active sites;
%e A362563    231 is {3},{1},{2}   -> NENENE, which has 4 active sites;
%e A362563    311 is {2,3},{},{1}  -> NNEENE, which has 3 active sites;
%e A362563    312 is {2},{3},{1}   -> NENENE, which has 4 active sites;
%e A362563    321 is {3},{2},{1}   -> NENENE, which has 4 active sites.
%e A362563 So T(3,2) = 1, T(3,3) = 3, T(3,4) = 4.
%Y A362563 Cf. A000079 (right diagonal), A000958 (1st column and row sums).
%K A362563 nonn,tabl
%O A362563 1,3
%A A362563 _Lara Pudwell_, Apr 24 2023