The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A197365 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A197365

T(n,k) gives the number of permutations of the set [n] that contain k occurrences of the subword (132); irregular array read by rows (n >= 0 and 0 <= k <= max(0, floor((n-1)/2))).
(history; published version)

#48 by Michael De Vlieger at Sat May 20 18:38:51 EDT 2023

COMMENTS

The recurrence follows from manipulation of the bivariate o.g.f/e.g.f. 1/W(u,z) = Sum_{n, k >= 0} T(n, k)*u^k*z^n/n!, whose reciprocal W(u,z) is the solution of the differential equation o.d.e. in Theorem 3.2 in Elizalde and Noy (2003) (with m = a = r + 1). The number t = r + 2 is the order of the differential equation o.d.e. in terms of the variable z.

KEYWORD

nonn,easy,tabf,changed

STATUS

proposed

approved

#47 by Jon E. Schoenfield at Fri May 19 18:19:50 EDT 2023

STATUS

editing

proposed

Discussion

Fri May 19

20:42

Jon E. Schoenfield: Please see discussion at A104997.

Sat May 20

15:04

N. J. A. Sloane: Better to stick with o.d.e.

18:38

Michael De Vlieger: Will revert this one.

#46 by Jon E. Schoenfield at Fri May 19 18:19:24 EDT 2023

COMMENTS

The recurrence follows from manipulation of the bivariate o.g.f/e.g.f. 1/W(u,z) = Sum_{n, k >= 0} T(n, k)*u^k*z^n/n!, whose reciprocal W(u,z) is the solution of the o.d.e. differential equation in Theorem 3.2 in Elizalde and Noy (2003) (with m = a = r + 1). The number t = r + 2 is the order of the o.d.e. differential equation in terms of the variable z.

STATUS

approved

editing

#45 by Michel Marcus at Wed Nov 06 03:35:27 EST 2019

STATUS

reviewed

approved

#44 by Joerg Arndt at Wed Nov 06 02:01:07 EST 2019

STATUS

proposed

reviewed

Discussion

Wed Nov 06

03:35

Michel Marcus: name:1243 keyword:tabf gives no hits

#43 by Petros Hadjicostas at Wed Nov 06 00:14:23 EST 2019

STATUS

editing

proposed

#42 by Petros Hadjicostas at Wed Nov 06 00:14:18 EST 2019

FORMULA

n-th row sum = n!. First column is A111004.

STATUS

proposed

editing

#41 by Petros Hadjicostas at Tue Nov 05 23:27:04 EST 2019

STATUS

editing

proposed

#40 by Petros Hadjicostas at Tue Nov 05 23:24:43 EST 2019

LINKS

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.

#39 by Petros Hadjicostas at Tue Nov 05 23:13:06 EST 2019

COMMENTS

(End)

Discussion

Tue Nov 05

23:15

Petros Hadjicostas: For the pattern 132 we have this irregular array, and for the pattern 12354 we have irregular array A264781. Where is the one for the pattern 1243?