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A342863 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1243. 0 <= k <= A028723(n + 1).

%I A342863 #19 Mar 28 2021 04:05:52
%S A342863 1,1,2,6,23,1,103,11,4,2,513,88,56,32,14,7,9,0,0,1,2761,638,543,341,
%T A342863 235,138,173,51,42,47,34,6,17,4,0,7,1,0,2,15767,4478,4600,3119,2658,
%U A342863 1710,2180,972,975,877,771,356,542,233,184,266,157,81,130,41,60,49,16,16,37,8,9,13,3,0,10,1,0,0,0,0,1
%N A342863 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1243. 0 <= k <= A028723(n + 1).
%C A342863 Equivalently the table for the patterns 2134, 3421, and 4312.
%C A342863 First column is A005802.
%H A342863 Peter Kagey, <a href="/A342863/b342863.txt">Rows n = 0..13, flattened</a>, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
%H A342863 Anders Kaseorg, <a href="https://codegolf.stackexchange.com/a/222494/53884">Answer: Patterns in Permutations</a>, Code Golf Stack Exchange.
%e A342863 Table begins:
%e A342863   n\k|       0        1        2        3        4        5        6
%e A342863   ---+-------------------------------------------------------------------
%e A342863    0 |       1;
%e A342863    1 |       1;
%e A342863    2 |       2;
%e A342863    3 |       6;
%e A342863    4 |      23,       1;
%e A342863    5 |     103,      11,       4,       2;
%e A342863    6 |     513,      88,      56,      32,      14,       7,       9, ...
%e A342863    7 |    2761,     638,     543,     341,     235,     138,     173, ...
%e A342863    8 |   15767,    4478,    4600,    3119,    2658,    1710,    2180, ...
%e A342863    9 |   94359,   31199,   36691,   26602,   25756,   17628,   22984, ...
%e A342863   10 |  586590,  218033,  284370,  218957,  231390,  166338,  221429, ...
%e A342863   11 | 3763290, 1535207, 2174352, 1767837, 1994176, 1496134, 2028316, ...
%Y A342863 Cf. A005802, A028723.
%Y A342863 Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342864 (1432), A342865 (1234).
%K A342863 nonn,tabf
%O A342863 0,3
%A A342863 _Peter Kagey_, Mar 26 2021