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In the OEIS you can find the coefficients of the h-polynomials for the associahedra, the Narayana polynomials A001263; for the stellahedra, A046802; and for the permutahedra, the Eulerian polynomials A008292. These h-polynomials as bivariate polynomials share the property that they are palindromic, i.e., the coefficients of the first half of the polynomials are the reverse, or a reflection of the second half; e.g., for the Eulerians

$E_4(a,b) = 1a^4 + 26a^3b + 66a^2b^2 + 26ab^3 + 1b^4.$

A second property, call it the $\Omega$ property, is that the absolute values of the coefficients of the first half of each polynomial (except for the linear polynomial) is increasing and therefore the second half, decreasing. (I believe this is true for all the h-polynomials above except for the first degree polynomials.)

I have rarely come across the converse situation, call it the U property, where there is a reflection symmetry but the first half is decreasing in absolute value rather than increasing. In fact, I can vaguely recall exploring such an OEIS entry long ago and have just come across a second example in the numerator polynomials of the power series expansion about $x=0$ of the square root of a quadratic equation; that is,

$$ \sqrt{1-x/z_1} \sqrt{1-x/z_2} = \sum_{n \geq 0} (-x)^{n}\sum_{k= 0 }^n \binom{1/2}{k} \binom{1/2}{n-k}(1/z_1)^k(1/z_2)^{n-k}$$

$= 1 - \frac{z_1 + z_2}{2 z_1 z_2}x - \frac{(z_1 - z_2)^2}{2^3 (z_1 z_2)^2}x^2 - \frac{(z_1 - z_2)^2 (z_1 + z_2)}{2^4 (z_1 z_2)^3}x^3 - \frac{(z_1 - z_2)^2 (5 z_1^2 + 6 z_2 z_1 + 5 z_2^2)}{2^7 (z_1 z_2)^4}x^4 - \frac{(z_1 - z_2)^2 (7 z_1^3 + 9 z_2 z_1^2 + 9 z_2^2 z_1 + 7 z_2^3)}{2^8 (z_1 z_2)^5}x^5 - \frac{(z_1 - z_2)^2 (21 z_1^4 + 28 z_2 z_1^3 + 30 z_2^2 z_1^2 + 28 z_2^3 z_1 + 21 z_2^4}{2^{10} (z_1 z_2)^6}x^6 + O(x^7) $

$= 1 - \frac{(z_1 + z_2)}{2 z_1 z_2)}x - \frac{z_1^2 - 2 z_2 z_1 + z_2^2}{2^3 (z_1 z_2)^2}x^2 - \frac{z_1^3 - z_2 z_1^2 - z_2^2 z_1 + z_2^3}{2^4 (z_1z_2)^3}x^3 - \frac{5 z_1^4 - 4 z_2 z_1^3 - 2 z_2^2 z_1^2 - 4 z_2^3 z_1 + 5 z_2^4}{2^7 (z_1z_2)^4}x^4 - \frac{7 z_1^5 - 5 z_2 z_1^4 - 2 z_2^2 z_1^3 - 2 z_2^3 z_1^2 - 5 z_2^4 z_1 + 7 z_2^5}{2^8 (z_1 z_2)^5}x^5 - \frac{21 z_1^6 - 14 z_2 z_1^5 - 5 z_2^2 z_1^4 - 4 z_2^3 z_1^3 - 5 z_2^4 z_1^2 - 14 z_2^5 z_1 + 21 z_2^6}{2^{10} (z_1 z_2)^6}x^6 + O(x^7).$

I haven't proven it, but I believe the U property holds for all the numerator polynomials of order four and higher. The first and last coefficients of the polynomials, the alleged highest in absolute value, are A098597 / A002596, related to the Catalan numbers, and the signed coefficients sum to zero for order 2 and above. With a change of variables these polynomials become essentially the Narayana polynomials.

I think one can write a script to search the OEIS database for examples of U polynomials, but I don't know how to do so. Can anyone provide examples from the OEIS (or elsewhere) of other U polynomial sequences, particularly those which belong to families sharing some combinatorial property, such as the h-polynomials above share?

Tom Copeland
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1 Answers1

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I wrote Java code that finds all OEIS sequences which, when interpreted as triangular arrays, are palindromic and U-shaped in each row. Most of them aren’t directly related to polynomials, but some specify generating functions (e.g. A078391, which is related to the Catalan numbers). Here’s the resulting list of links:

A001231 A003984 A010032 A027420 A048147 A048149 A049581 A049723 A049735 A049747 A049751 A051125 A059366 A067804 A068555 A073254 A074911 A078391 A080382 A083457 A084867 A085484 A085517 A098361 A101037 A103451 A103516 A108063 A114929 A114972 A115253 A115255 A116588 A129821 A131816 A131821 A131843 A131844 A131919 A134634 A140685 A143180 A143182 A143183 A144155 A144216 A144217 A154869 A155169 A155720 A155725 A156003 A156047 A156130 A157050 A157927 A161419 A169894 A171824 A173049 A173424 A173476 A176081 A176862 A181262 A183632 A183642 A183652 A183662 A183672 A183680 A183918 A183986 A184003 A184029 A184039 A184048 A184386 A184457 A184477 A184505 A186609 A186619 A187645 A198063 A198064 A198065 A203994 A204004 A204008 A204022 A204123 A204143 A204154 A204156 A204158 A204237 A204284 A204361 A204370 A204381 A204408 A204417 A204492 A204501 A204607 A204754 A206772 A209727 A209898 A224651 A224658 A224806 A227074 A227075 A227076 A227550 A227581 A231205 A233997 A234140 A234227 A234266 A234490 A234549 A234564 A234658 A234665 A234672 A234681 A234690 A234702 A234712 A234728 A234738 A234823 A234867 A235017 A235026 A235087 A235098 A235107 A235175 A235186 A235198 A235212 A235239 A235248 A235258 A235280 A235289 A235301 A235310 A235319 A235884 A235893 A235954 A237015 A237637 A239287 A244027 A250742 A251128 A251149 A251236 A252254 A253044 A253468 A255228 A257361 A257447 A258316 A258612 A258894 A258921 A258957 A259222 A259704 A259770 A267089 A271709 A271710 A283845 A284128 A284131 A298594 A298597 A302339 A302346 A317773 A318354 A321126 A323621 A331886 A338032 A338796 A346500 A346517 A346538 A347563 A349039 A349203 A350770 A364480 A367108

joriki
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  • Would be nice to find the subset of the list above of those entries that describe a list of polynomials $P_n(x) = \sum_{k=0}^n a_{n,k} x^k$ for which $a_{n,0} = a_{n,n} \neq 0$ for $n$ say greater than $4$ . Another possibility is for OEIS entries for which $a_{n,0} =0$ and $a_{n,1} = a_{n,n} \neq 0$ and the U property holds for the non-zero coefficients. (Give a guy an inch and he'll take a mile.) – Tom Copeland Dec 29 '23 at 00:02