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 A263484 Triangle read by rows: T(n,k) (n>=1, 0<=k
 1, 1, 1, 1, 2, 3, 1, 3, 7, 13, 1, 4, 12, 32, 71, 1, 5, 18, 58, 177, 461, 1, 6, 25, 92, 327, 1142, 3447, 1, 7, 33, 135, 531, 2109, 8411, 29093, 1, 8, 42, 188, 800, 3440, 15366, 69692, 273343, 1, 9, 52, 252, 1146, 5226, 24892, 125316, 642581, 2829325 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums give A000142, n >= 1. From Allan C. Wechsler, Jun 14 2019 (Start): Suppose we are permuting the numbers from 1 through 5. For example, consider the permutation (1,2,3,4,5) -> (3,1,2,5,4). Notice that there is exactly one point where we can cut this permutation into two consecutive pieces in such a way that no item is permuted from one piece to the other, namely (3,1,2 | 5,4). This "cut" has the property that all the indices to its left are less than all the indices to its right. There are no other such cut-points: (3,1 | 2,5,4) doesn't work, for example, because 3 > 2. Stanley defines the "connectivity set" as the set of positions at which you can make such a cut. In this case, the connectivity set is {3}. In the present sequence, T(n,k) is the number of permutations of n elements with k cut points. (End) Essentially the same triangle as [1, 0, 0, 0, 0, 0, 0, 0, …] DELTA [0, 1, 2, 2, 3, 3, 4, 4, 5, …] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 18 2020 LINKS Alois P. Heinz, Rows n = 0..150, flattened FindStat - Combinatorial Statistic Finder, The cardinality of the complement of the connectivity set. Math Stack Exchange, Discussion of this sequence, June 2019. Richard P. Stanley, The Descent Set and Connectivity Set of a Permutation, arXiv:math/0507224 [math.CO], 2005. EXAMPLE Triangle begins:   1,   1, 1,   1, 2,  3,   1, 3,  7, 13,   1, 4, 12, 32,  71,   1, 5, 18, 58, 177, 461,   ... Triangle [1, 0, 0, 0, 0, ...] DELTA [0, 1, 2, 2, 3, 3, ...] :   1;   1, 0;   1, 1,  0;   1, 2,  3,  0;   1, 3,  7, 13,  0;   1, 4, 12, 32, 71, 0; ... - Philippe Deléham, Feb 18 2020 MATHEMATICA rows = 11; (* DELTA is defined in A084938 *) Most /@ DELTA[Table[Boole[n == 1], {n, rows}], Join[{0, 1}, LinearRecurrence[{1, 1, -1}, {2, 2, 3}, rows]], rows] // Flatten (* Jean-François Alcover, Feb 18 2020, after Philippe Deléham *) PROG (Sage Math) # cf. FindStat link def statistic(x):      return len(set(x.reduced_word())) for n in [1..6]:     for pi in Permutations(n):         print(pi, "=>", statistic(pi)) CROSSREFS Cf. A000142. T(n,n-1) gives A003319. A version with reflected rows is A059438, A085771. T(2n,n) gives A308650. Sequence in context: A011117 A069269 A193092 * A293985 A100324 A121424 Adjacent sequences:  A263481 A263482 A263483 * A263485 A263486 A263487 KEYWORD nonn,tabl AUTHOR Christian Stump, Oct 19 2015 EXTENSIONS More terms from Fred Lunnon and Christian Stump. Name changed by Georg Fischer as proposed by Allan C. Wechsler, Jun 13 2019. STATUS approved

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Last modified September 21 20:10 EDT 2021. Contains 347598 sequences. (Running on oeis4.)