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A342862 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 2143. 0 <= k <= A028723(n + 1).

%I A342862 #22 Mar 28 2021 04:05:43
%S A342862 1,1,2,6,23,1,103,11,4,2,513,88,53,33,18,8,6,0,0,1,2761,642,495,340,
%T A342862 262,160,172,65,58,39,14,6,18,0,0,6,0,0,2,15767,4567,4099,3007,2692,
%U A342862 1832,2171,1152,1291,968,728,457,566,174,176,221,129,14,122,29,38,52,8,0,32,9,0,10,0,0,8,0,0,0,0,0,1
%N A342862 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 2143. 0 <= k <= A028723(n + 1).
%C A342862 Equivalently the table for the pattern 3412.
%C A342862 First column is A005802.
%H A342862 Peter Kagey, <a href="/A342862/b342862.txt">Rows n = 0..13, flattened</a>, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
%H A342862 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000407">St000407: The number of occurrences of the pattern 2143 in a permutation</a>.
%H A342862 Anders Kaseorg, <a href="https://codegolf.stackexchange.com/a/222494/53884">Answer: Patterns in Permutations</a>, Code Golf Stack Exchange.
%e A342862 Triangle begins:
%e A342862   n\k|       0        1        2        3        4        5        6
%e A342862   ---+-------------------------------------------------------------------
%e A342862    0 |       1;
%e A342862    1 |       1;
%e A342862    2 |       2;
%e A342862    3 |       6;
%e A342862    4 |      23,       1;
%e A342862    5 |     103,      11,       4,       2;
%e A342862    6 |     513,      88,      53,      33,      18,       8,       6, ...
%e A342862    7 |    2761,     642,     495,     340,     262,     160,     172, ...
%e A342862    8 |   15767,    4567,    4099,    3007,    2692,    1832,    2171, ...
%e A342862    9 |   94359,   32443,   32345,   25049,   24492,   17732,   21841, ...
%e A342862   10 |  586590,  232189,  250371,  203452,  211291,  160561,  201524, ...
%e A342862   11 | 3763290, 1679295, 1926145, 1635315, 1776655, 1409304, 1787218, ...
%Y A342862 Cf. A005802, A028723.
%Y A342862 Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342863 (1243), A342864 (1432), A342865 (1234).
%K A342862 nonn,tabf
%O A342862 0,3
%A A342862 _Peter Kagey_, Mar 26 2021