|
%I #33 Aug 12 2022 20:18:12
%S 1,1,2,6,23,1,103,10,6,1,512,77,69,30,21,5,6,2740,548,598,330,335,123,
%T 174,58,58,37,26,3,9,1,15485,3799,4686,2970,3411,1676,2338,1040,1317,
%U 878,777,363,608,230,252,165,133,30,93,26,31,4,1,3,4,91245,26165,35148,24550,30182,17185,24685,12976,16867,12248,12360,7203,11086,5692,6391,5194,5006,2751,3917,2019,2482,1622,1371,812,1233,490,495,416,360,157,282,54,78,41,29,22,49,7,4,0,6
%N Irregular triangle: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 4213. 0 <= k <= A342646(n).
%C The sequence is the same for the patterns 1342, 2431, and 3124.
%C The sequence appears to be the same for the patterns 1423, 2314, 3241, and 4132.
%C First column is given by A022558. Row sums given by n!.
%H Peter Kagey, <a href="/A342840/b342840.txt">Rows n = 0..13, flattened</a>, based on Anders Kaseorg's Rust program at the Code Golf Stack Exchange link.
%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000750">St000750: The number of occurrences of the pattern 4213 in a permutation.</a>
%H Anders Kaseorg, <a href="https://codegolf.stackexchange.com/a/222494/53884">Answer: Patterns in Permutations</a>, Code Golf Stack Exchange.
%H Rob Pratt, <a href="https://math.stackexchange.com/a/4071338/121988">Greatest number of occurrences of the pattern 4213 in a permutation</a>, Mathematics Stack Exchange.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationPattern.html">Permutation Pattern</a>
%e Triangle begins:
%e n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12 13
%e ----+-------------------------------------------------------------
%e 0 | 1;
%e 1 | 1;
%e 2 | 2;
%e 3 | 6;
%e 4 | 23, 1;
%e 5 | 103, 10, 6, 1;
%e 6 | 512, 77, 69, 30, 21, 5, 6;
%e 7 | 2740, 548, 598, 330, 335, 123, 174, 58, 58, 37, 26, 3, 9, 1;
%t Join@@Array[Table[Length@Select[Permutations@Range@#,Length@Select[Subsets[#,{4}],Ordering@Ordering@#=={4,2,1,3}&]==k&],{k,0,Binomial[n+1,4]}]//.{a__,0}:>{a}&,8,0] (* _Giorgos Kalogeropoulos_, Mar 25 2021 *)
%Y Cf. A022558, A342646.
%Y Cf. A263771 (analogous for 312).
%K nonn,tabf
%O 0,3
%A _Peter Kagey_, Mar 24 2021
|
