This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A327722 #67 Feb 16 2025 08:33:58
%S A327722 1,1,1,1,1,2,1,1,2,5,1,1,2,6,16,1,1,2,6,23,63,1,1,2,6,24,110,296,1,1,
%T A327722 2,6,24,119,630,1623,1,1,2,6,24,120,708,4204,10176,1,1,2,6,24,120,719,
%U A327722 4914,32054,71793,1,1,2,6,24,120,720,5026,38976,274914,562848
%N A327722 Number T(m,n) of permutations of [n] avoiding the consecutive pattern 12...(m+1)(m+3)(m+2), where m, n >= 0; array read by ascending antidiagonals.
%C A327722 By taking complements of permutations, we see that T(m,n) is also the number of permutations of [n] avoiding the consecutive pattern (m+3)(m+2)...(3)(1)(2). [The complement of permutation (c_1,c_2,...,c_n) of [n] is (n + 1 - c_1, n + 1 - c_2, ..., n + 1 - c_n).]
%C A327722 If we let S(n,k) = T(n-k, k) for n >= 0 and 0 <= k <= n, we get a triangular array shown in the Example section below.
%C A327722 Note that lim_{n -> oo} S(n,k) = k! = A000142(k) for k >= 0.
%C A327722 By using the ratio test and the Stirling approximation to the Gamma function, we may show that the radius of convergence of the power series W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(b(n, m+2)*((m + 2)*n + 1)) is infinity (for each m >= 0). Thus, the function W_m(z) (as defined by the power series) is entire.
%H A327722 A. Baxter, B. Nakamura, and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.html">Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes</a>, 2011.
%H A327722 Sergi Elizalde and Marc Noy, <a href="https://doi.org/10.1016/S0196-8858(02)00527-4 ">Consecutive patterns in permutations</a>, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116) with u = 0 and m and a in the theorem equal to our m + 1.
%H A327722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%H A327722 Wikipedia, <a href="https://en.wikipedia.org/wiki/Falling_and_rising_factorials">Falling and rising factorials</a>.
%F A327722 E.g.f for row m >= 0: 1/W_m(z), where W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(b(n, m+2)*((m + 2)*n + 1)) with b(n, k) = A329070(n, k) = (k*n)!/(k^n * (1/k)_n). (Here (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol, or rising factorial, which is denoted by (x)^n in some papers and books.)
%F A327722 The function W_m(z) satisfies the o.d.e. W_m^(m+2)(z) + z*W_m'(z) = 0 with W_m(0) = 1, W_m'(0) = -1, and W_m^(s)(0) = 0 for s = 2..(m + 1).
%F A327722 T(m, n) = Sum_{s = 0..floor((n - 1)/(m + 2))} (-(m + 2))^s * (1/(m + 2))_s * binomial(n, (m + 2)*s + 1) * T(m, n - (m + 2)*s - 1) for n >= 1 with T(m, 0) = 1.
%F A327722 T(m, n) = n! for 0 <= n <= m + 2.
%F A327722 T(m, m+3) = (m + 3)! - 1 = A000142(m + 3) - 1 = A033312(m + 3) for m >= 0. [In the set of permutations of [m + 3] there is exactly one permutation that contains the pattern 12...(m+1)(m+3)(m+2).]
%F A327722 Conjecture: T(m, m + 4) = A242569(m + 4) = (m + 4)! - 2*(m + 4) for m >= 0.
%F A327722 Limit_{m -> oo} T(m, n) = n! = A000142(n) for n >= 0.
%e A327722 Array T(m, n) (with rows m >= 0 and columns n >= 0) begins as follows:
%e A327722 1, 1, 2, 5, 16, 63, 296, 1623, 10176, 71793, ...
%e A327722 1, 1, 2, 6, 23, 110, 630, 4204, 32054, 274914, ...
%e A327722 1, 1, 2, 6, 24, 119, 708, 4914, 38976, 347765, ...
%e A327722 1, 1, 2, 6, 24, 120, 719, 5026, 40152, 360864, ...
%e A327722 1, 1, 2, 6, 24, 120, 720, 5039, 40304, 362664, ...
%e A327722 1, 1, 2, 6, 24, 120, 720, 5040, 40319, 362862, ...
%e A327722 ...
%e A327722 Triangular array S(n, k) = T(n-k, k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e A327722 1;
%e A327722 1, 1;
%e A327722 1, 1, 2;
%e A327722 1, 1, 2, 5;
%e A327722 1, 1, 2, 6, 16;
%e A327722 1, 1, 2, 6, 23, 63;
%e A327722 1, 1, 2, 6, 24, 110, 296;
%e A327722 1, 1, 2, 6, 24, 119, 630, 1623;
%e A327722 1, 1, 2, 6, 24, 120, 708, 4204, 10176;
%e A327722 1, 1, 2, 6, 24, 120, 719, 4914, 32054, 71793;
%e A327722 1, 1, 2, 6, 24, 120, 720, 5026, 38976, 274914, 562848;
%e A327722 ...
%Y A327722 Rows include A111004 (m = 0, pattern 132), A117226 (m = 1, pattern 1243), A202213 (m = 2, pattern 12354).
%Y A327722 Cf. A000142, A033312, A242569, A329070.
%K A327722 nonn,tabl
%O A327722 0,6
%A A327722 _Petros Hadjicostas_, Nov 02 2019